ELLIPTIC CURVES AND ICOSAHEDRAL GALOIS REPRESENTATIONSegoins/notes/thesis.pdf · 2012. 7. 3. · L-series and its functional equation. In the 1930’s, Emil Artin [Art65] conjectured

ELLIPTIC CURVES AND

ICOSAHEDRAL GALOIS REPRESENTATIONS

a dissertation

submitted to the department of mathematics

and the committee on graduate studies

of stanford university

in partial fulfillment of the requirements

for the degree of

doctor of philosophy

Edray Herber Goins

October 2002

c© Copyright by Edray Herber Goins 2003

All Rights Reserved

ii

I certify that I have read this dissertation and that, in

my opinion, it is fully adequate in scope and quality as a

dissertation for the degree of Doctor of Philosophy.

Daniel W. Bump(Principal Adviser)




Karl C. Rubin




David B. Carlton

Approved for the University Committee on Graduate

Studies:

iii

Introduction

0.1 Artin’s Conjecture

Conjecture 0.1.1. Let ρ : Gal(F a/F )→ GLn(C) be an irreducible nontrivial repre-

sentation with finite projective image, where F is a number field. Then the associated

L-series L(ρ, s) is an entire function.

In 1917, Erich Hecke [Hec87] proved a series of results about certain characters

which are now commonly referred to as Hecke characters; one corollary states that

one-dimensional complex Galois representations give rise to entire L-series. He re-

vealed, through a series of lectures [Hec83] at Princeton’s Institute for Advanced

Study in the years that followed, the relationship between such representations as

generalizations of Dirichlet characters and modular forms as the eigenfunctions of

a set of commuting self-adjoint operators. Many mathematicians were inspired by

his ground-breaking insight and novel proof of both the analytic continuation of the

L-series and its functional equation.

In the 1930’s, Emil Artin [Art65] conjectured that a generalization of such a

result should be true; that is, irreducible complex projective representations of finite

Galois groups should also give rise to entire L-series. He came to this conclusion

after proving himself that both 3-dimensional and 4-dimensional representations of

the simple group of order 60, the alternating group A5, give rise to L-series with

singularities. In the spirit of Hecke, he phrased his conjecture in terms of both the

analytic continuation of the L-series and its functional equation. It is known, due to

the insight of Robert Langlands [Lan80b] in the 1970’s relating Hecke characters with

Representation Theory, that in order to prove the conjecture it suffices to prove that

iv

such representations are automorphic. This conjecture and has been the motivation

for much study in both Algebraic and Analytic Number Theory ever since.

The case of one-dimensional Galois representations was proved in full generality

with the advent of Class Field Theory. Indeed, any one-dimensional Galois represen-

tation ρ : Gal(F a/F )→ C× must necessarily be abelian, so that by Artin Reciprocity

ρ can be associated with a character χ defined on the multiplicative group of ade-

les A×F/F

×. That is, every one-dimensional representation may be associated with a

Hecke character, so that Hecke’s 1917 result implies the validity of Artin’s Conjecture

for n = 1.

Many mathematicians, inspired by this result, began work on the irreducible two-

dimensional representations. Felix Klein [Kle93] had showed that the only finite

images in PGL2(C) correspond to the Platonic Solids. That is, they are the rota-

tions of the polygons (Zn or Dn), the tetrahedron (A4), the octahedron (S4), and

the icosahedron (A5). It suffices to consider irreducible two-dimensional projective

representations with the aforementioned images in order to prove Artin’s Conjecture

for n = 2.

Most of these cases of the conjecture have been answered in the affirmative. Ir-

reducible cyclic and dihedral representations (that is, representations whose image

in PGL2(C) is isomorphic to Zn or Dn, respectively) may be interpreted as repre-

sentations induced from abelian ones, so that the proof of analytic continuation may

be reduced to one using Hecke characters. Irreducible tetrahedral and some octahe-

dral representations (i.e. projective image isomorphic to A4 or S4, respectively) were

proved to give entire L-series due to work by Langlands [Lan80a] in the 1970’s on base

change forGL(2). The remaining cases for irreducible octahedral representations were

proved shortly thereafter by Jerrold Tunnell [Tun81]. Such methods worked because

they exploited the existence proper nontrivial normal subgroups. Unfortunately, the

simple group of order 60 has none, so it is still not known whether the irreducible

icosahedral representations (i.e. projective image isomorphic to A5) have analytic

continuation. The first known example to verify Artin’s conjecture in this case did

not surface until Joe Buhler’s work [Buh78] in 1977.

For general n-dimensional representations, not much is known. It is easy to show

v

that the L-series constructed from an arbitrary representation is analytic in the right-

half plane Re(s) > 1. In 1947, Richard Brauer [Bra47] proved that the characters

associated to representations of finite groups are a finite linear combination of one-

dimensional characters, and so the corresponding L-series have meromorphic continu-

ation; that is, the functions have at worst poles at a finite number of places. Brauer’s

proof does not guarantee that the integral coefficients of such a linear combination

are positive; it can be shown that in many cases the coefficients are negative so that

the proof of continuation to the entire complex plane may be reduced to showing that

the poles of the L-series are cancelled by the zeroes.

0.2 Recent Results

One approach to the icosahedral case of the Artin Conjecture starts by realizing an

icosahedral Galois extension K/F as one which is contained in the field generated

by the 5-torsion of an abelian variety of dimension 1. This idea is motivated by the

isomorphism A5∼= PSL2(F5). It is well known that any Galois group isomorphic

A5 is the splitting field of a quintic. Klein [Kle93] showed that after adjoining some√α and

√5 such a splitting field may be realized as that extension generated by the

x-coordinates of the 5-torsion points from a suitable elliptic curve. The L-series of the

complex representation is congruent the L-series of that elliptic curve modulo some

prime ideal lying above 5; in fact, the complex representation is a Galois deformation

of the 5-adic representation. Using the recent work of Andrew Wiles [Wil95], Fred

Diamond [Dia96], and Richard Taylor [Tay97] along with formulations of conjectures

by Barry Mazur [MT90], one may hope to use the automorphicity of elliptic curves

over number fields to prove automorphicity of icosahedral representations. This ap-

proach has the drawback that the elliptic curve must be defined over a biquadratic

extension F(√

α,√

5).

A slightly different approach has been taken by Nick Shepherd-Barron and Richard

Taylor [SBT97]. Considering the isomorphism A5∼= SL2(F4), they noted that icosa-

hedral extensions may be realized as the field generated by the x-coordinates of the

2-torsion of an abelian variety of dimension 2. This approach has the benefit that the

vi

surface is defined over the same field F as the representation. One must still quote

results of Wiles-Diamond-Taylor with the conjectures of Mazur in order to hope to

deduce the automorphicity of such representations; the difference is that one only

need work over the ground field F instead of some extension.

This dissertation seeks to approach the icosahedral case of the Artin Conjecture

by combining the classical work of Klein with modern results on elliptic curves. We

worked with a specific icosahedral representation defined over Q and an elliptic curve

defined over Q(√

5) with a encouraging property: The twist of the 5-adic representa-

tion by a Hecke character is associated to the base change of modular form of weight

2 defined over Q. That is, this representation is a deformation of the original icosa-

hedral representation; base change to an extension is not necessary. This approach

has the promise that the abelian surface is a familiar elliptic curve, and we can still

work over the original field of definition for the representation.

0.3 Approach of This Dissertation

This thesis is concerned with the modular icosahedral representation constructed by

Buhler in [Buh78]. The overall goal of this thesis is to show how such a representation

is attached to an elliptic curve. This was to be accomplished through several tasks:

Task #1: To attach an elliptic curve EB via [Kle93] to the quintic used in [Buh78],

and show explicitly how the splitting field of this quintic is related to the field

generated by the x-coordinates of the 5-torsion points on this curve.

Task #2: To show that the elliptic curve EB is modular, even though the curve may

be defined over some extension of Q(√

5).

Task #3: To describe how the modular form of weight 1 described in [Buh78] is

associated with the modular form of weight 2 attached to EB.

Buhler describes his icosahedral extension K/Q explicitly as the splitting field of a

quintic defined over Q. Using the work of Klein, we found an elliptic curve EB defined

vii

over Q(√

5) such that K(ζ5) = Q(EB[5]x) is the field generated by the x-coordinates

of the 5-torsion.

Upon further investigation, we found that EB is 2-isogenous to a twist of its Galois

conjugate. The elliptic curve EB was shown be shown to be modular (i.e. attached

to a Hilbert modular form for Q(√

5) ) by a novel method: there is a modular form

fχ over Q of nontrivial nebentype such that, once one base changes to Q(√

5) and

twists by a Hecke character of order 4, the traces of Frobenius match. (A result of

Gerd Faltings [Fal83] shows that one only has to check enough traces to guarantee

modularity. We used a result of Ron Livne [Liv87] to explictly find a bound on the

number.) The representation ρf associated with this modular form is a deformation

of the complex representation ρB found in [Buh78].

Unfortunately, while we know that ρf and ρB are congruent to each other modulo

a prime ideal lying above 5, we have not been able to prove that ρB is modular.

If we were able to do derive this fact from the modularity of EB, we could derive

the results in [Buh78]. Most of the current results on deformations (most notably

Diamond’s [Dia96] generalizations of the results in Wiles’ [Wil95]) assume that the

abelian variety is semistable. Our elliptic curve has additive reduction at all primes

above 2 and 5, so we could not use these ideas.

Nonetheless, the methods outlined in this thesis can be used as an attack for

Artin’s Conjecture: Assume that ρ : Gal(F a/F )→ GL2(C) an icosahedral represen-

tation defined over a number field F . We have shown that there is a family of elliptic

curves E defined over a quadratic extension of F (√

5) such that, for some Hecke char-

acter χ, the twisted mod 5 representation χ ⊗ ρE,5 is isomorphic to the base change

of ρ. Artin’s Conjecture should follow from the automorphicity of elliptic curves over

number fields if we could show that the mod 5 representation is automorphic when

the 5-adic representation is automorphic for some elliptic curve in this family.

The encouraging promise of our methods relies on the realization that our par-

ticular elliptic curve EB is a Q-curve. (The definition of a Q-curve can be found in

[Rib92].) We conjecture that when F = Q, a Q-curve always lurks in the picture. As-

suming that this is the case, we have shown that for some Hecke character, the twisted

5-adic representation — not just the mod 5 representation — is the base change of

viii

a weight 2 representation over F . This representation would be a deformation of

ρ. Recent work of Ken Ribet [Rib92], Ki-ichiro Hashimoto [Has99], Yuji Hasagewa

[Has97], Josep Gonzalez, and Joan Lario [GL98] strongly suggest that a proof of the

conjecture that all Q-curves are indeed modular will soon be complete. We conjecture

that Artin’s Conjecture for icosahedral representations over Q will indeed follow from

deformations of mod 5 representations coming from Q-curves.

The specific layout of this work is described in the next section.

ix

Layout of Thesis

Chapter 1. An overview of elliptic curves is given, with derivation of the 2-, 3-, 4-,

and 5-division polynomials. It is meant to give both background and motivation

for the remainder of the thesis.

Chapter 2. Here the reader will find a modern treatment of Klein’s work on the

Icosahedron. We present an algorithm to construct an elliptic curve having

relevant properties with the 5-torsion points once a quintic polynomial is given.

We work over fields of characteristic not dividing 60.

Chapter 3. This chapter contains generalizations for the Platonic Solids, where we

present a similar algorithm to construct elliptic curves having relevant properties

with N -torsion points once a polynomial of degree less than 6 is given. We work

over fields of characteristic not dividing the order of PSL2 (Z/N Z) where N is

2, 3, 4, or 5.

Chapter 4. Here definitions about complex and `-adic representations are given

along with definitions of Artin L-series and L-series attached to elliptic curves.

We work over number fields.

Chapter 5. Congruence relations are established between L-series coming from com-

plex Galois representations and L-series coming from elliptic curves. We show

that any representation of Platonic Type is the twist of the base change of the

mod N representation of an elliptic curve. We continue to work over number

fields.

x

Chapter 6. This chapter contains the detailed study of the icosahedral representa-

tion found in [Buh78]. An elliptic curve is constructed via the results in chapter

2, and its properties are studied in detail. The curve is shown to be a modular

Q-curve.

Chapter 7. We conclude the thesis by discussing the role of Q-curves as a possible

future approach.

xi

Acknowledgments

The author would like to thank his advisor, Dan Bump, for suggesting the original

problem; his unofficial co-advisor Karl Rubin for his invaluable suggestions and the

realization that the elliptic curve is isogenous to its conjugate; and graduate student

William Stein for his help in calculating coefficients via Cremona’s modular symbol

algorithm. This thesis could not have been completed without their help.

The author would also like to thank the following persons for helpful conversations:

David Carlton, Jordan Ellenberg, Ralph Greenburg, Kenneth Ribet, and Richard

Taylor. Papers from Ki-ichiro Hashimoto and Yuji Hasegawa were also helpful in

gaining insight into the properties of Q-curves.

All of the research was sponsored by a generous fellowship from the National

Physical Science Consortium (NSPC) and the National Security Agency (NSA).

The author would like to give ultimate credit to his mother, Eddi Beatrice Goins,

and her godfather, William Herber Dailey, for constant guidance, support, and pro-

tection over the years.

xii

List of Symbols

ρ an irreducible complex Galois representation

ρ the projectivization of ρ

im(ρ) the image of ρ

ker(ρ) the kernel of ρ

χ a Hecke character

ωN a Dirichlet character modulo N

fρ the (Artin) conductor of ρ

F a number field

K a finite extension of F , the kernel of ρ

L an extension of K, the kernel of ρ

F a a separable algebraic closure of the field F

GF the profinite group Gal(F a/F )

σ, τ elements of the Galois group

OF the ring of integers in F

p, P prime ideals of F and L respectively

q(x) a polynomial of degree N defined over F

Disc(q) the discriminant of q(x)

Tsch(q) the Tschirnhaus discriminant of q(x)

xiii

θ(x) a polynomial of degree N with no xN−1 and xN−2 terms

E an elliptic curve

P , Q points on E

E(K) collection of points on E contained in K

j(E), j0 j-invariant

` a fixed prime

N a power of `

E[N ] the N -torsion points on E defined over F a

F (j0) the smallest field of definition for an elliptic curve E with j(E) = j0

F0 that extension of F (j0) generated by the N -th roots of unity ζN

F (E[N ]) the field generated by the coordinates of the N -torsion on E

F (E[N ]x) the subfield generated by just the x-coordinates

fE the conductor of E

ρE,` the `-adic representation of E

ρE, N the mod N representation of E

ψN the N -division polynomial

θE, N a monic principal polynomial of degree N with roots in F (E[N ]x)

ζN a primitive Nth root of unity

Z/N Z the ring of N elements

Zn the cyclic group of order n

Dn the dihedral group of order 2n

Sn the symmetric group on n letters

An the alternating group on n letters

GL(V ) the general linear group of a vector space V

SL(V ) the special linear group

Z the center of GL(V )

xiv

Z diagonal matrices of GL(V ) with determinant ±1

PGL(V ) the projective general linear group

PSL(V ) the projective special linear group

ϕN the representation Z · SL2 (Z/N Z)→ GL2(C)

µ, ν indices

ε twice the real part of ζ5 or −1+√

52

=j0(z) the polynomial defining the icosahedral equation

EB the elliptic curve y2 = x3 + (5−√

5)x2 +√

5x

χB a specific Hecke character of order 4

εB the unique Dirichlet character modulo 4

εχ the tensor product εB ⊗ ω5

ρB the icosahedral representation studied in [Buh78]

fB the modular form in S1 (Γ0(800), εB) attached to ρB

fχ a modular form in S2 (Γ0(800), εχ)

ρf the representation attached to fχ which is a deformation of ρB

xv

Contents

Introduction iv

0.1 Artin’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

0.2 Recent Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

0.3 Approach of This Dissertation . . . . . . . . . . . . . . . . . . . . . . vii

Layout of Thesis x

Acknowledgments xii

List of Symbols xiii

1 Torsion on Elliptic Curves 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Points of Order 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Points of Order 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Points of Order 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Points of Order 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Lectures on The Icosahedron 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Geometric Realization of A5 . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Invariant Forms and Principal Quintics . . . . . . . . . . . . . . . . . 16

2.4 The Icosahedral Equation . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 5-Division Points on Elliptic Curves . . . . . . . . . . . . . . . . . . . 24

xvi

3 The Platonic Solids 30

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Geometric Realization of PSL2 (Z/N Z) . . . . . . . . . . . . . . . . 30

3.3 Invariant Forms and Principal Polynomials . . . . . . . . . . . . . . . 36

3.4 N -Division Points on Elliptic Curves . . . . . . . . . . . . . . . . . . 42

4 Representations and L-Series 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 The Frobenius Element . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 L-Series Attached to Galois Representations . . . . . . . . . . . . . . 51

4.4 `-adic Representations from Elliptic Curves . . . . . . . . . . . . . . . 55

5 Representations of Platonic Type 59

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Associated Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Complex Representations of Z · SL2 (Z/N Z) . . . . . . . . . . . . . . 62

5.4 Complex Representations from Elliptic Curves . . . . . . . . . . . . . 65

5.5 Congruences Among L-Series . . . . . . . . . . . . . . . . . . . . . . 68

6 Deformations of Buhler’s Modular Form 71

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.2 Principal Quintics and Elliptic Curves . . . . . . . . . . . . . . . . . 72

6.3 The Elliptic Curve EB . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.4 The Hecke Character χB . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.5 Base Change and Modularity . . . . . . . . . . . . . . . . . . . . . . 81

6.6 The Icosahedral Representation ρB . . . . . . . . . . . . . . . . . . . 85

7 Final Remarks 88

A Tables and Data 90

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

A.2 Basic Properties of Q(√

5) . . . . . . . . . . . . . . . . . . . . . . . . 91

A.3 Structure of Gal(Q(√

5){2,5}/Q(√

5))

. . . . . . . . . . . . . . . . . . 93

xvii

A.4 Group Structure of EB(Fp) . . . . . . . . . . . . . . . . . . . . . . . . 95

A.5 Supersingular Primes of EB . . . . . . . . . . . . . . . . . . . . . . . 97

A.6 Coefficients of the L-Series . . . . . . . . . . . . . . . . . . . . . . . . 98

Bibliography 103

xviii

Chapter 1

Torsion on Elliptic Curves

1.1 Introduction

We now review the basic facts of elliptic curves which will be used in subsequent

chapters. The reader who is familiar with these facts is urged to skip the following

chapter. This chapter is included to fix notation to be used in the sequel as well as

to list some obscure facts about points of finite order. Some of the results below can

be found in Serre [Ser72].

Let F be a field, and let E(F ) be the collection of points in F , including the point

at infinity O, satisfying the cubic equation

y2 + a1 x y + a3 y = x3 + a2 x2 + a4 x+ a6 (1.1)

where the coefficients are all in F . (In practice, we will choose F to be a finite

extension of the rational numbers, and later we will choose F = Q.) With this curve

come the classically associated quantities

1

CHAPTER 1. TORSION ON ELLIPTIC CURVES 2

b2 = a21 + 4 a2

b4 = 2 a4 + a1 a3

b6 = a23 + 4 a6

b8 = a21 a6 + 4 a2 a6 − a1 a3 a4 + a2 a

23 − a2

4

c4 = b22 − 24 b4

c6 = −b32 + 36 b2 b4 − 216 b6

∆ = −b22 b8 − 8 b34 − 27 b26 + 9 b2 b4 b6

(1.2)

We say that E is an elliptic curve defined over F if and only if the the discriminant

∆ of the curve is invertible. The j-invariant of an elliptic curve is defined as the ratio

j(E) = c34/∆.

The points on an elliptic curve form an abelian group by the chord-tangent con-

struction. Let P and Q be two such points different from O. Since E may be defined

by a cubic equation, any line through two rational points intersects the curve in a

third, say P ∗ Q. (If P = Q, we choose this line to be the line tangent to the curve

at P .) Set (x, y) ∗ O = (x, −y − a1 x − a3), and define addition of two points as

P ⊕ Q = (P ∗ Q) ∗ O. It is clear that P ⊕ Q = Q ⊕ P since the choice of the

initial point for drawing the line is unimportant. Also, P ⊕ O = P , so that O is

the additive identity, and [−1]P = P ∗ O is the additive inverse. Though it is quite

tedious to prove, this choice of addition is associative. For a more intuitive feel of

this construction, the reader may consult Silverman and Tate [ST92].

For any positive integer N , define [N ]P = P ⊕P ⊕· · ·⊕P as addition of the point

N times. The goal of this chapter is to study those points P of E(F a) defined over a

separable algebraic closure of F satisfying [N ]P = O, i.e. those points of order N .

1.2 Points of Order 2

For this section, assume that F is a field with characteristic different from 2.

A point P has order 2 if and only if P = [−1]P . That is, (x, y) = (x, −y−a1 x−a3)

so y = −(a1 x+a3)/2. Substituting this value for y into the original cubic curve yields

roots of the polynomial


ψ2(x) = 4x3 + b2 x2 + 2 b4 x+ b6 = (2 y + a1 x+ a3)

2 (1.3)

We refer to this as the 2-division polynomial. It is straightforward to check that

the discriminant of this polynomial is 16 ∆, and that the roots are given by

xν =1

12

(−b2 +

3

√c6 +

√c26 − c34 ζν

3 +3

√c6 −

√c26 − c34 ζ2ν

3

), ν = 1, 2, 3.

(1.4)

where ζ3 is a primitive cube root of unity. This expression is valid only when the

ground field F has characteristic different from both 2 and 3.

Proposition 1.2.1 (2-Division). Let F be a field of characteristic different from 2,

E be an elliptic curve defined over F , and ψ2 be the 2-division polynomial as in (1.3).

Set F (E[2]x) as its splitting field, and G = Gal(F (E[2]x)/F ).

1. If ψ2 is irreducible and ∆ is not a square, then G ∼= S3.

2. If ψ2 is irreducible and ∆ is a square, then G ∼= Z3.

3. If ψ2 is reducible and ∆ is not a square, then G ∼= Z2.

4. Otherwise G is trivial.

Proof. ψ2 is a cubic, so the Galois group is contained in S3. The statement follows

by considering whether the Galois group has an element of order 2 or 3.

Consider the case when the 2-division polynomial ψ2 is irreducible and the discrim-

inant ∆ is not a square. We will explicitly list the automorphisms of the Galois group.

Let S be that automorphism of order 2 that fixes x1, and T be that automorphism

of order 2 that fixes x3. Then

Gal(F (E[2]x)/F ) =⟨S, T | S2 = T 2 = (TS)3 = 1

⟩ ∼= S3 (1.5)

One verifies that F (xν) is the fixed field of the subgroup 〈S (TS)ν〉, and K(√

∆) is

the fixed field of the subgroup 〈TS〉.


We may also identify this Galois group as a collection of matrices. For the three

roots xν , let Pν = (xν ,−(a1 xν + a3)/2). The discriminant of ψ2 is assumed to be

nonzero, so we have exactly three distinct points. Adding in the identity O, it is easy

to see that E[2] ∼= Z/2Z ⊕ Z/2Z. We expect Gal(F (E[2]x)/F ) to be the same as

Aut(E[2]) ∼= GL2(Z/2Z). Indeed this is the case: Choose the basis P1 = (1, 0) and

P2 = (0, 1) so that P3 = P1 ⊕ P2 = (1, 1). The cycle S takes P3 7→ P2 and P2 7→ P3,

and the cycle T takes P1 7→ P2 and P2 7→ P1. This induces the matrix representations

S 7→

(1 1

1

), T 7→

(1

1

)as matrices in GL2 (Z/2Z). (1.6)

That is, we have the identifications

Gal(F (E[2]x)/F ) ∼=

⟨(1 1

1

),

(1

1

)⟩= GL2 (Z/2Z) ∼= S3 (1.7)

Hence calculating the Galois group is equivalent to calculating the group of auto-

morphisms of E[2]. (When ψ2 is reducible we have inclusion rather than an isomor-

phism.)

The converse of proposition 1.2.1 is also true.

Corollary 1.2.2 (S3 Extensions). Let F be a field of characteristic different from

2, and K/F be a Galois extension with group S3. Then there exists an elliptic curve

E/F such that K = F (E[2]x).

Proof. Write Gal(K/F ) = 〈σ, τ | σ3 = τ 2 = 1, τ σ τ = σ−1〉, and let Kτ be the

fixed field of 〈τ〉 so that [Kτ : F ] = 3. Choose q1 ∈ Kτ − F , and let q(x) be its

minimal polynomial. Clearly q(x) is a monic cubic with coefficients in F . We show

that the equation y2 = q(x) defines an elliptic curve E with the desired properties.

The discriminant Disc(q) is nonzero since the roots of q(x) are the distinct conjugates

qν = σνq1; that is, E is indeed an elliptic curve.

Clearly F (E[2]x) ⊆ K is the splitting field of q(x), so we show equality by

considering the degree of F (E[2]x)/F . By construction the 2-division polynomial

ψ2(x) = 4 · q(x) is irreducible, and its discriminant ∆ = 16 Disc(q) is not a square or


else Kτ would be Galois implying 〈τ〉 is a normal subgroup of S3. Proposition 1.2.1

states [F (E[2]x) : F ] = 6, so we must have equality: K = F (E[2]x).

This corollary is the motivation for the next two chapters. We will show that if

K/F is a field extension — with suitable restrictions on the characteristic of F —

such that the Galois group is either A4, S4, or A5, then there is an elliptic curve such

that K is contained in F (E[N ]x) for N being either 3, 4, or 5. The curious reader

may consult theorem 3.4.1.


For this section, assume that F is a field with characteristic different from 3.

A point P has order 3 if and only if [2]P = [−1]P . Considering the x-coordinates

we have

x4 − b4 x2 − 2 b6 x− b84x3 + b2 x2 + 2 b4 x+ b6

= x2 P = x−P = x (1.8)

That is, we seek the roots of the 3-division polynomial

ψ3(x) = 3x4 + b2 x3 + 3 b4 x

2 + 3 b6 x+ b8 (1.9)

We present a couple of ways to see that ψ3 has no repeated roots. It is easy

to check that ddxψ3(x) = 3ψ2(x), so any repeated root would also be a root of ψ2.

This would generate a 2-division point, not a 3-division point. Alternatively, the

discriminant of ψ3 is Disc (ψ3) = −33 ∆2. The discriminant of the elliptic curve ∆ is

nonzero by assumption so the discriminant of the polynomial is nonzero as well.


E be an elliptic curve defined over F , and ψ3 be the 3-division polynomial as in (1.9).

Set F (E[3]x) as its splitting field, and G = Gal(F (E[3]x)/F (ζ3) ).

1. If ψ3 is irreducible over F (ζ3) and ∆ is not a cube, then G ∼= A4.

2. If ψ3 is irreducible and ∆ is a cube, then G ∼= D2.


3. If ψ3 is reducible and ∆ is not a cube, then G ∼= Z3.

4. Otherwise G is contained in S2.

Proof. We consider the resolvent cubic of this quartic. Say that we have the factor-

ization ψ3(x) = 3∏

ν(x− xν), and consider the roots

θ1 = b4 − 3 (x1x2 + x3x4) , θ2 = b4 − 3 (x1x3 + x2x4) , θ3 = b4 − 3 (x1x4 + x2x3) .

(1.10)

Gal(F (E[3]x)/F ) permutes these θν among themselves, so they must be roots of

some cubic polynomial θE,3(x) =∏

ν(x − θν) with coefficients in F . Expressing the

polynomial in terms of the original coefficients we find that θE,3(x) = x3 −∆. This

shows that both 3√

∆ and ζ3 are contained in the splitting field so that the extension

F (E[3]x)/F (ζ3) is normal.

In order to prove the theorem, we note that ψ3 is a degree 4 polynomial whose

discriminant Disc (ψ3) = −33 ∆2 is a square in F (ζ3). The Galois group is really

contained in A4 = D2 o Z3, which has order 12 = 4 × 3. The irreducibility of ψ3

contributes the factor of 4 while the irreducibility of θE,3 contributes the factor of

3. The theorem becomes immediate once one considers the possible subgroups of

A4.

From this discussion, we find that the roots of ψ3 are explicitly given by

xµ =1

12

(−b2 +

3∑ν=1

(−1)bµ ν2c

√c4 + ζν−1

33

√c26 − c34

), µ = 1, 2, 3, 4; (1.11)

where ζ3 is a primitive cube root of unity and b·c is the greatest integer function.

This expression is valid only when the characteristic of F is different from 2 and 3.

In order to find the y-coordinates, we use the original equation for the elliptic

curve as in (1.1). For each root xµ, the discriminant of this equation — when viewed

as a quadratic in y — is ψ2(xµ) = 13

ddxψ3(xµ) 6= 0. Hence this equation has two


distinct roots y for each x-coordinate xµ, so upon adding the point at infinity, we see

that there are exactly nine points of order 3. That is, E[3] ∼= Z/3Z⊕ Z/3Z.

We present another way to view the theorem above. Denote F (E[3]) as the field

containing both the x- and y-coordinates of all the 3-division points. Any auto-

morphism σ from Gal(F (E[3]) /F (ζ3) ) permutes the roots of the polynomial ψ3;

in particular, σ([3]P ) = [3] (σP ) so σ permutes the points in E[3]. Choose a basis

for E[3], so that σ may be written as a matrix in GL2(Z/3Z). The discriminant

Disc(ψ3) =∏

ν<µ(xµ−xν)2 = −33 ∆2 is a square over F (ζ3), so this matrix has deter-

minant 1. (Another way to see this is to consider the Weil pairing; consult [Sil86].) In

other words, Gal(F (E[3]) /F (ζ3) ) must be contained in SL2(Z/3Z). When restrict-

ing to the action on the x-coordinates, the automorphism P 7→ −P acts trivially, and

so

Gal(F (E[3]x) /F (ζ3) ) ⊆ PSL2(Z/3Z) ∼= A4 (1.12)

We will return to points of order 3 in chapter 3.


For this section, assume that F is a field with characteristic prime to 4.

A point P has order 4 if and only if [2]P is a point of order 2. Set x2 P is its

x-coordinate; we want to know when it is a root of ψ2. Consider the expression

x2 P =x4 − b4 x2 − 2 b6 x− b84x3 + b2 x2 + 2 b4 x+ b6

=⇒ ψ2(x2 P ) =ψ4(x)

ψ2(x)3(1.13)

where we have defined the following as the 4-division polynomial

ψ4(x) = 2x6 + b2 x5

+ 5 b4 x4 + 10 b6 x

3 + 10 b8 x2 + (b2 b8 − b4 b6)x+ (b4 b8 − b26)

(1.14)

(The usual definition of the 4-division polynomial which the reader may be more


familiar with is the product of 2 y + a1 x+ a3 with ψ4.)

The roots of this polynomial will yield points of exact order 4, while the roots of

ψ2 yields points of exact order 2. Recall that ψ2 has discriminant 16 ∆ and so it has

distinct roots. On the other hand ψ4 has discriminant Disc (ψ4) = −44 ∆5 and so it

has distinct roots as well.

We wish to describe the Galois group of this polynomial in more detail. To this

end, we establish the following lemma.

Lemma 1.4.1. Let F be a field of characteristic different from 2, E be an elliptic

curve defined over F , and F (E[2]x) be the splitting field of the 2-division polynomial

ψ2 defined in (1.3). Given that ρν are the roots of ψ2, there is the factorization

ψ4(x) = 23∏

ν=1

[(x− ρν)

2 − ψ′2(ρν)

4

]over F (E[2]x). (1.15)

Moreover, the points P = (x, y) of exact order 4 have x- and y-coordinates

xµν = ρν + (−1)µ√

(ρν − ρν−1) (ρν − ρν+1)

yµν = −a1 xµν + a3

2± (xµν − ρν)

[√ρν − ρν−1 + (−1)µ√ρν − ρν+1

] (1.16)

and the fourth root 4√

∆ may be expressed in terms of these coordinates.

The fact that 4√

∆ may be expressed in terms of x− and y− coordinates of the 4-

division points is stated but not proved in Serre [Ser72]. One checks that 4√

∆ cannot

be expressed in terms of the x-coordinates alone.

Proof. ψ2(x2 P ) = ψ4(x)/ψ2(x)3, so consider the equivalent relation x2 P = ρ as a root

of ψ2(x). This implies the equation

0 =(x4 − b4 x2 − 2 b6 x− b8

)− ρ

(4x3 + b2 x

2 + 2 b4 x+ b6)

=

[(x− ρ)2 − 12 ρ2 + 2 b2 ρ+ 2 b4

4

]2

=

[(x− ρ)2 − 1

4ψ′2(ρ)

]2 (1.17)


By considering degrees and the leading coefficient, the statement about the factor-

ization follows.

Using the factorization ψ2(x) = 4∏

ν(x − ρν), we find that the roots of the 4-

division polynomial are xµν = ρν + (−1)µ√

(ρν − ρν−1) (ρν − ρν+1). For each root x,

recall that y is given by the relation 2 y + a1 x+ a3 = ±√ψ2(x). Using the relation

ψ2(xµν) = 4∏

ρ

(xµν − ρ) = 4 (xµν − ρν)2[√ρν − ρν−1 + (−1)µ√ρν − ρν+1

]2(1.18)

we find that the points of exact order 4 are in the form stated above.

To prove the last statement about 4√

∆, denote y∗µν as the corresponding yµν with

the “±” sign reversed. Then we have the expression yµν − y∗µν = ±√ψ2(xµν) which

gives [1

128

∏ν

∑µ

(−1)µyµν − y∗µν

xµν − ρν

]4

=1

16

∏ν

(ρν − ρν+1)2 = ∆ (1.19)

This completes the proof.


E be an elliptic curve defined over F , ψ4 be the 4-division polynomial as in (1.14),

and denote θE,4 as the polynomial

θE,4(x) = x4 +32 ∆x+4 c4 ∆ with resolvent θ∗E,4(x) = x3− c4 ∆x+16 ∆2. (1.20)

Set F (E[4]x) as its splitting field of ψ4, and G = Gal(F (E[4]x)/F (ζ4) ).

1. If both θE,4 and θ∗E,4 are irreducible over F (ζ4) and ∆ is not a square, then

G ∼= S4.

2. If both θE,4 and θ∗E,4 are irreducible but ∆ is a square, then G ∼= A4.

3. If θE,4 is irreducible but θ∗E,4 is reducible, then either G ∼= D4 or G ∼= Z4.


4. Otherwise G is contained in S3.

Note the relation here with 3-division points. In proposition 1.3.1 we considered

the polynomial θE,3(x) = x3−∆ to see if the Galois group had a 3-cycle. In this case,

the fourth root 4√

∆ is not contained properly in F (E[4]x) so we cannot introduce

the polynomial x4−∆, but we can introduce the polynomial θE,4 to see if the Galois

group has a 4-cycle.

Proof. First we show that the Galois group G is contained in S4 by showing that

F (E[4]x) is the splitting field of a quartic. The field F (E[4]x) is generated by the

x-coordinates xµν as in lemma 1.4.1, and the quartic θE,4(x) =∏

µ(x− θµ) as defined

in (1.20) has the roots θµ =∑

ν(−1)bµν2c√−∆/(x0ν − ρν), where b·c is the greatest

integer function. Hence, the θµ also generate F (E[4]x), so it is indeed the splitting

field of a quartic.

The Galois group G is equal to S4 if and only if G has a 4-, 3-, and 2-cycle.

Information about the splitting field of a quartic polynomial is well-known; it depends

on the irreducibility of the quartic θE,4, its resolvent cubic θ∗E,4, and the discriminant

Disc(θE,4) = 47 c26 ∆3. The reader may verify the details on a case by case basis.

As before, we may interpret the proposition by considering the action of the

Galois group Gal(F (E[4])/F (ζ4) ) on E[4]. Corresponding to the six x-coordinates

as described in lemma 1.4.1, we have twelve points of exact order 4; while including

the three points of exact order 2, we have a total of 16 points of order dividing 4.

Hence E[4] ∼= Z/4Z⊕Z/4Z. Any automorphism σ of Gal(F (E[4])/F (ζ4) ) permutes

these elements, so σ ∈ GL2(Z/4 Z). The matrices must have determinant 1 due to

consideration of the Weil pairing. (The Galois group is restricted to a field extension

which contains the fourth roots of unity.) Considering the action on just the x-

coordinates,

Gal(F (E[4]x) /F (ζ4) ) ⊆ PSL2(Z/4Z) ∼= S4 (1.21)

The quartic polynomial θE,4 defined in (1.20) is a member of a larger family of

quartics which generate S4-extensions. We will return to points of order 4 in chapter


3 where we will derive this family and study its properties.


For this final section, assume that F is a field with characteristic different from 5.

A point P has order 5 if and only if [4]P = [−1]P . After a bit of algebra, we

consider the x-coordinates to find the polynomial ψ5 = ψ22 ψ4 − ψ3

3. Explicitly,

ψ5(x) = 5x12 + 5 b2 x11 + (b22 + 31 b4)x

10 + 5 (2 b2 b4 + 19 b6)x9

+ 15 (3 b2 b6 + 7 b8)x8 + 30 (b4 b6 + 3 b2 b8)x

7

+ 15 (−b2 b4 b6 − b26 + b22 b8 + 10 b4 b8)x6

+ (−b22 b4 b6 − 51 b2 b26 + b32 b8 + 46 b2 b4 b8 + 174 b6 b8)x

5

+ 5 (−b22 b26 − 12 b4 b26 + b22 b4 b8 + 11 b2 b6 b8 − 25 b28)x

4

+ 5 (−2 b2 b4 b26 − 5 b36 + 2 b22 b6 b8 + 2 b4 b6 b8 − 7 b2 b

28)x

3

+ 5 (−2 b2 b36 + 2 b2 b4 b6 b8 + 3 b26 b8 − 5 b4 b

28)x

2

+ 5 b6 (−b4 b26 + b2 b6 b8 − 5 b28)x+ (b4 b26 b8 − b46 − b38)

(1.22)

We refer to this as the 5-division polynomial. It can be found by considering the

formula for x2P , and then composing it with itself to find x4P . We then consider the

roots of x4P = x.

The discriminant is Disc(ψ5) = 511 ∆22 so that it has distinct roots. Again, the

y-coordinates are given by a quadratic equation with discriminant ψ2(xµ), so by

the same reasoning as before we find 24 distinct points of exact order 5 and the

isomorphism E[5] ∼= Z/5Z ⊕ Z/5Z. As a result, we may consider the action by the

Galois group on just the x-coordinates — with consideration of the Weil pairing —

to see that

Gal(F (E[5]x) /F (ζ5) ) ⊆ PSL2(Z/5Z) ∼= A5 (1.23)

It is clear that when the 5-division polynomial ψ5 is irreducible, 12 divides the


order of the Galois group. We will find in the next chapter that the splitting field of

the principal quintic

θE,5(x) = x5 − 40 ∆x2 − 5 c4 ∆x− c24 ∆ (1.24)

so that we have a 5-cycle if and only if this quintic is irreducible.

The next proposition explains how the splitting fields of these two polynomials

are related.


E be an elliptic curve defined over F , ψ5 be the 5-division polynomial as in (1.22),

and θE,5 be the principal quintic as in (1.24).

Set F (E[5]x) as the splitting field of ψ5, and F (θE,5) as the splitting field of θE,5.

The field F (θE,5) is contained in F (E[5]x), and if both ψ5 and θE,5 are irreducible

then we have the following structure as Galois groups:

Gal(F (θE,5) /F (√

5) ) ∼= Gal(F (E[5]x) /F (ζ5)) ∼= A5 (1.25)

The this proposition will be the focus of the next chapter; it is just a restatement

of theorem 2.4.1. Points of order 5 will be studied in more detail in chapter 2 where we

will present a method to attach elliptic curves to quintic polynomials that generate A5

extensions. We will find that while icosahedral extensions come from consideration

of the extension F (E[5]x)/F (ζ5), it is not necessary to work over a ground field

containing the fifth roots of unity. Indeed, the extension F (θE,5)/F (√

5) works just

as well, and has the added bonus that the ground field is at worst a quadratic extension

of F .

Chapter 2

Lectures on The Icosahedron

2.1 Introduction

In this chapter, we work entirely with points of order 5 to discover properties of

quintic polynomials as they are related to elliptic curves. Most of the exposition that

follows is motivated by Felix Klein’s seminal work [Kle93] on the icosahedron, with

appropriate generalizations and clarifications in the results from that text.

2.2 Geometric Realization of A5

The icosahedron is a Platonic solid which consists of 12 vertices, 30 edges (where 5

surround each vertex), and 20 faces. In this section, we describe several irreducible

representations of rotations of the icosahedron, including the 3-dimensional and the

projective 2-dimensional. To this end, we begin our discussion by explicitly listing

the vertices as points on the unit sphere:

P (±)ν =

(± 1√

5, ± 2√

5cos

2πν

5,

2√5

sin2πν

5

)for ν ∈ Z/5Z; P (±)

∞ = (±1, 0, 0).

(2.1)

There are 60 rotations which bring the icosahedron into itself; it is well known that

this group is the alternating group on 5 letters, A5. One generator S of this group

13

CHAPTER 2. LECTURES ON THE ICOSAHEDRON 14

has order 5; it keeps the z-axis fixed, but cycles the points P(±)ν 7→ P

(±)ν±1. Another

generator is a transposition T that maps P(±)∞ 7→ P

(±)0 , P

(±)1 7→ P

(±)4 , and P2 7→ P

(∓)3 .

Yet another is an involution U that maps P(±)ν 7→ P

(∓)ν .

As matrices relative to the choice of points in (2.1), these generators may be

represented as

S =

1

cos 2π5− sin 2π

5

sin 2π5

cos 2π5

T =

1√5

2√5

2√5− 1√

5

−1

U =

−1

−1

1

(2.2)

This is an irreducible 3-dimensional representation of A5, of which there are two

possible. The other comes about by interchanging the representations for S2 and S.

The group A5 is generated by these three cycles S, T , and U . It is easy to

check the relations U T = T U and USU−1 = S−1; and that there are the subgroups

D2 = 〈T, U〉, D5 = 〈S, U〉, and A4 = 〈S4TS2, T, U〉. (Dn is the dihedral group of

order 2n, and An is the alternating group on n letters.) If the reader prefers cycle

notation, set S = [1 2 3 4 5], T = [1 2] [3 4], and U = [1 5] [2 4]; then the previously

mentioned relations and subgroups may be verified directly. We leave it for the

reader to verify that U can be expressed in terms of S and T .

There is a conformal equivalence $ of the unit sphere with the complex projective

line P1(C) — commonly called Stereographic Projection — given by the canonical

maps

(x0, x1, x2)$7−→ x1 + ı x2

1− x0

;z1

z2

$−1

7−→(|z1|2 − |z2|2

|z1|2 + |z2|2,

2 Re(z1 z2)

|z1|2 + |z2|2,

2 Im(z1 z2)

|z1|2 + |z2|2

).

(2.3)

(Recall that the real line R may be identified with the equator and the upper half

plane in C with the upper half of the sphere.) Under stereographic projection, the

points on the icosahedron transform as


$P±ν =

1±√

5

2ζ±ν5 for ν ∈ Z/5Z, $ P+

∞ =∞, $ P−∞ = 0. (2.4)

This equivalence $ gives us a canonical isomorphism with fractional linear trans-

formations and A5.

Proposition 2.2.1 (Projective A5 Representations). Let {P (±)ν } be the collection

of vertices of the icosahedron as in (2.1), $ be stereographic projection as in (2.3),

and ς : {P (±)ν } → P1(F5) be the map P

(±)ν 7→ ±ν which identifies antipodal points and

maps to the subscripts.

Given an automorphism σ ∈ A5 = 〈S, T, U〉, the induced map σ$ = $ ◦ σ ◦$−1

is a fractional linear transformation in PGL2(C). Explicitly,

S$ z = ζ5 z, T$ z =z + ε

ε z − 1, U$ z = −1

z; (2.5)

where ζ5 is a primitive fifth root of unity, and ε = ζ5 + ζ−15 .

Similarly, the induced map σς = ς ◦ σ ◦ ς−1 is a fractional linear transformation

in PSL2(F5). Explicitly,

Sς ν ≡ ν + 1 (mod 5), T ς ν ≡ −1

ν(mod 5), U ς ν ≡ −ν (mod 5). (2.6)

In particular, A5 may be embedded in PGL2(C) as a projective 2-dimensional repre-

sentation, and A5∼= PSL2(F5).

The reader may simplify the proposition above with the following diagram:

P1(C)$←−−− {P (±)

ν } ς−−−→ P1(F5)

PGL2(C)

y A5

y yPSL2(F5)

P1(C)$−1

−−−→ {P (±)ν } ς−1

←−−− P1(F5)

(2.7)

Proof. Any automorphism σ ∈ A5 corresponds to an induced map σ$ = $ ◦ σ ◦$−1

as a fractional linear transformation. If we identify fractional linear transformations


as matrices in the projective linear group, then we have an explicit representation of

A5 as a subgroup of PGL2(C).

Now consider the canonical mapping from the upper half of the sphere ς given

by ς : P(+)ν 7→ ν. Any automorphism σ ∈ A5 corresponds to an induced map

σς = ς ◦ σ ◦ ς−1. Again, if we identify fractional linear transformations as matrices

in the projective linear group, then we have an explicit isomorphism of A5 with

PSL2(F5).

2.3 Invariant Forms and Principal Quintics

For this section, we let F be a field of characteristic relatively prime to 60; that is,

the characteristic is not 2, 3, or 5. We will show how to embed the splitting field of

a quintic polynomial in the splitting field of a certain quintic attached to an elliptic

curve. The work in this section is a paraphrase of the results in [Kle93].

Define the following homogeneous polynomials:

∆(z1, z2) =

(−z1 z2

4∏ν=0

[z1 −$P (+)

ν z2

] [z1 −$P (−)

ν z2

])5

(2.8)

c4(z1, z2) =(z201 + z20

2

)− 228

(z151 z

52 − z5

1z152

)+ 494 z10

1 z102 (2.9)

c6(z1, z2) =(z301 + z30

2

)+ 522

(z251 z

52 − z5

1z252

)− 10005

(z201 z

102 + z10

1 z202

)(2.10)

where $P(±)ν are defined as in (2.4).

These polynomials satisfy the relation c34−c26 = 1728 ∆. The polynomial ∆(z1, z2)

vanishes on the 12 vertices of the (projected) icosahedron, while c4(z1, z2) vanishes on

the midpoints of the 20 faces and c6(z1, z2) vanishes on the midpoints of the 30 edges.

Considering these forms as polynomials in one variable, they have the discriminants

Disc(∆1/5) = 525, Disc(c4) = 260 310 595, and Disc(c6) = −290 360 5205; which shows

quite explicitly that they have distinct roots as long at the characteristic of F does


not divide 60.

One verifies that each form is invariant under A5 = 〈S$, T$, U$〉. In the next

chapter we will give a proof of this fact and a derivation of this class of polynomials.

Still, the anxious reader may verify the invariance of these forms by checking that

this is indeed the case for the two generators S$ and T$.

While these three invariant forms are homogeneous polynomials, they are not ho-

mogeneous of degree 0. ∆(z1, z2) is invariant under A5, so the polynomial ∆(z1, z2)1/5

has nontrivial action by S$ but is otherwise invariant under D2 = 〈T$, U$〉. (Recall

that the action by S$ is just multiplication by ζ5; see (2.5).) Consider the homoge-

neous rational function

r(z1, z2) =[z2

1 + z22 ]

2[z21 − 2 ·$P (+)

0 z1 z2 − z22

]2 [z21 − 2 ·$P (−)

0 z1 z2 − z22

]2∆(z1, z2)1/5

(2.11)

This form is a well-defined rational function on P1(F a). It vanishes at the fixed

points of T$, U$, and T$ U$ — this explains where the quadratic factors in the

numerator come from — and has poles at the vertices of the icosahedron as in (2.4).

The zeroes of r + 3 are zeroes of c4, and the zeroes of r2 + 10 r + 45 are zeroes of c6.

(However, the converse is not true; not all zeroes of c4 are zeroes of r + 3, etc.)

Since r(z1, z2) has nontrivial action by a 5-cycle S$, we may associate a polynomial

of degree 5 to this rational function. The key insight due to Klein is that we may

associate a principal quintic i.e. a degree 5 polynomial θ(x) with the x4 and x3 terms

missing.

Lemma 2.3.1 (Klein). Let F be a field of characteristic different from 5, and r =

r(z1, z2) be the rational function defined in (2.11). Then for arbitrary constants m

and n in the algebraic closure F a, the forms

θν =m

rν + 3+

n

(rν + 3) (r2ν + 10 rν + 45)

where rν = r ◦ (S$)ν ; (2.12)

are roots of a principal quintic


θ(x) =∏

ν∈Z/5 Z

(x− θν) = x5 + Ax2 +B x+ C (2.13)

with coefficients

A = −20

j

[(2m3 + 3m2 n

)+ 432

6mn2 + n3

1728− j

]

B = −5

j

[m4 − 864

3m2 n2 + 2mn3

1728− j+ 559872

n4

(1728− j)2

]

C = −1

j

[m5 − 1440

m3 n2

1728− j+ 62208

15mn4 + 4n5

(1728− j)2

](2.14)

where j = c34/∆ = (r + 3)3 (r2 + 11 r + 64).

Note that the terms in (2.14) are each homogeneous rational functions of degree 0

in z1 and z2. In the next section, the rational function j = j(z1, z2) will be associated

with an elliptic curve. The fact that it is invariant under A5 will allow us to associate

it with the j-invariant.

Proof. This can be verified directly using a symbolic calculator. (The reader may

also consult [Kle93], though Klein assumes one is working over C.)

Every principal quintic arises in this way:

Lemma 2.3.2 (Klein). Let F be a field of characteristic different from 2 and 5, and

let θ(x) = x5 +Ax2 +B x+C be a principal quintic defined over F . Then there exist

m, n, and j in the quadratic extension F (√

5Disc(θ)) such that system (2.14) holds.

Proof. By eliminating the variables n and j in system (2.14), we find that m is a root

of the quadratic equation


(A4 − 5B3 + 25AB C)m2 + (11A3B + 50B2C − 125AC2)m

+ (64A2B2 − 135A3C − 125B C2) = 0

(2.15)

which has the discriminant 5 Disc(θ) in terms of

Disc(θ) = −27A4B2 + 256B5

+ 108A5C − 1600AB3C + 2250A2B C2 + 3125C4(2.16)

The other variables n and j may be expressed as rational functions of m:

n =1

12

A2m4 + 10ABm3 + 45 (B2 − 2AC)m2 − 90BC m− 675C2

(5AC −B2)m+ 5BC(2.17)

j = − 5

A2

(Am2 + 3Bm− 15C)3

(5AC −B2)m+ 5BC(2.18)

The result follows.

Although the formulas in the proof above can be found in [Kle93], Klein assumes

the discriminant Disc(θ) is a square so that m, n, and j are defined over F (√

5). One

can work through Klein’s formulas to verify that in general m, n, and j are indeed

defined over the quadratic extension F (√

5 Disc(θ)).

Klein was well aware that any quintic polynomial could be put into principal form,

so working over this class of quintics is not as restrictive as one may expect. One

accomplishes this by performing a Tschirnhaus transformation:

Lemma 2.3.3 (Tschirnhaus). Through a quadratic substitution, every quintic can

be transformed into a principal quintic.

That is, let F be a field of characteristic different from 2 and 5, and q(x) be a

quintic defined over F with roots qν. Then there exist α, β, and γ in some quadratic


extension F(√

Tsch(q))

such that the quintic polynomial defined over this quadratic

extension, with roots α q2ν + β qν + γ, has its x4 and x3 terms missing.

Proof. Say that q(x) =∑

µ aµ xµ = a5

∏ν(x − qν), and form the resolvent quintic

polynomial θ(x) =∏

ν(x − θν) with roots θν = α q2ν + β qν + γ for some α, β, and γ

to be found. To set the x4 and x3 terms to zero, we define α, β, and γ as

α = 5 a5

(2 a2

4 − 5 a3 a5

)

β =5 (4 a3

4 − 13 a3 a4 a5 + 15 a2 a25) + 5 a5

√Tsch(q)

2

γ =5 (a3 a

24 − 4 a2

3 a5 + 3 a2 a4 a5) + a4

√Tsch(q)

2

(2.19)

where the Tschirnhaus discriminant is defined as

Tsch(q) = 5(−3 a2

3 a24 + 8 a2 a

34

+ 12 a33 a5 − 38 a2 a3 a4 a5 + 16 a1 a

24 a5 + 45 a2

2 a25 − 40 a1 a3 a

25

) (2.20)

Hence θ(x) = x5 +Ax2 +B x+C has coefficients in the extension F (√

Tsch(q) ).

Putting these three lemmas together, we arrive at the key result of this section.

Theorem 2.3.4. Let F be a field of characteristic different from 2 and 5, q(x) be

a quintic defined over F , and denote K as its splitting field. There exists j0 in the

biquadratic extension

F(√

Tsch(q),√

5Disc(q))

(2.21)

with Tsch(q) as given in (2.20), such that for any elliptic curve E in Weierstrass

form with j(E) = j0, K(j0) is the splitting field of both


(r+ 3)3 (r2 + 11 r+ 64) = j0 and θE,5(x) = x5 − 40 ∆x2 − 5 c4 ∆x− c24 ∆. (2.22)

Proof. Let q(x) =∏

ν(x− qν), and choose α, β, and γ as in the proof of lemma 2.3.3

so that θν = α q2ν +β qν +γ are the roots of a principal quintic θ(x). Following lemma

2.3.2 and (2.12), choose m, n, and j0 such that

α q2ν + β qν + γ = θν =

m

rν + 3+

n

(rν + 3) (r2ν + 10 rν + 45)

(2.23)

where rν = r ◦ (S$)ν and j0 = (r+3)3 (r2 +11 r+64). Hence, the roots of q(x) are in

one-to-one correspondence with the roots of the quintic equation (r+ 3)3 (r2 + 11 r+

64) = j0.

In particular, this latter quintic in r does not depend on m and n — as long as

m and n are constants in the same field extension as j0. Given any elliptic curve E

with j(E) = j0, form the principal quintic as in (2.12) with m = c4 and n = 0:

θE,5(x) =∏ν

(x− c4

rν + 3

)= x5 − 40 ∆x2 − 5 c4 ∆x− c24 ∆ (2.24)

Hence, the roots of θE,5(x) are in one-to-one correspondence with the roots of the

quintic (r + 3)3 (r2 + 11 r + 64) = j0, so the statement in the proposition follows.

It remains to show that j0 is contained in biquadratic extension F ′ generated by√Tsch(q) and

√5 Disc(q). From lemmas 2.3.2 and 2.3.3, it is clear that j0 is defined

over a quadratic extension of F (√

Tsch(q)) generated by√

5 Disc(θ). The relation

Disc(θ) =

[∏ν<µ

((qν + qµ) +

β

α

)]2

·Disc(q) (2.25)

shows that F ′ is also the field generated by√

Tsch(q) and√

5 Disc(θ), so that we

have equality.

θE,5(x) is the quintic mentioned in (1.24). We will study more of its properties in

the following sections.


2.4 The Icosahedral Equation

Given a quintic polynomial q(x) defined over F , there is an associated quantity j0

defined over the algebraic closure F a. Conversely, given j0 in F a, we can solve the

equation j0 = c34/∆ for ξ = z1/z2. We define the icosahedral equation as =j0(z) = 0,

where

=j0(z) = c4(z, 1)3 − j0 ∆(z, 1)

=

[(z20 + 1

)− 228

(z15 − z5

)+ 494 z10

]3

+ j0

[z5(z10 + 11 z5 − 1

)5](2.26)

with discriminant Disc(=j0) = 5785 j400 (j0 − 1728)30, so that the icosahedral equation

has distinct roots if the characteristic of F is different from 5. Once we solve this

equation for a root ξ, we find r(ξ, 1) and then θν(ξ, 1) by (2.12). The roots of q(x)

may be expressed in terms of θν(ξ, 1) as described in the proof of 2.3.4.

The following theorem states the explicit relationship between the splitting fields

of q(x), θ(x), and =j0(z).

Theorem 2.4.1 (A5 Extensions - Algebraic Form). Let F be a field of charac-

teristic prime to 60, and let q(x) be a quintic polynomial over F with Galois group

A5. There exists j0 in the biquadratic extension F (√

5,√

Tsch(q)), with Tsch(q) as

given in (2.20), such that for any elliptic curve E with j(E) = j0,

splitting field

of q(x)⊆

splitting field

of θE,5(x)⊆

splitting field

of =j0(z)(2.27)

More precisely, set F0 = F (ζ5,√

Tsch(q)), and denote the splitting field of q(x)

as K, the splitting field of θE,5(x) as F (θE,5), and the splitting field of =j0(z) as K0.

Then F (θE,5) = K · F (j0), K0 = K · F0 and we have the isomorphisms

Gal(K/F ) ∼= Gal(K(j0)/F (j0) ) ∼= Gal(K0/F0 ) ∼= A5 (2.28)


The drawback to this result is that the splitting field K does not always equal

the splitting field K0 of the icosahedral equation. The best we can hope to do is

compromise on the intermediate extension F (θE,5) = K(j0) which is a biquadratic

extension of K. This is the underlying theme of this chapter.

Proof. Given q(x), let j0 be as in Theorem 2.3.4. By assumption, the discriminant

Disc(q) is a square, so j0 lies in the biquadratic extension stated above. We discuss

how the splitting fields of q(x), θE,5, and =j0(z) are related to each other.

By the First Isomorphism Theorem for groups,

Gal(K/K ∩ F (j0) ) ∼= Gal(K · F (j0)/F (j0) ), Gal(K/K ∩ F0) ∼= Gal(K · F0/F0);

(2.29)

so in order to prove the theorem we show that K ∩F0 = K ∩F (j0) = F , K ·F (j0) =

F (θE,5) and K · F0 = K0. (For a statement of the Isomorphism Theorems, consult

[DF91].)

K ∩ F0/F an extension of degree dividing 8, so Gal(K/K ∩ F0) is a subgroup of

Gal(K/F ) = A5 of index dividing 8. A5 has no nontrivial subgroups of this type. (An

has no nontrivial subgroups of index less than n when n ≥ 5.) Hence K ∩ F0 = F ,

and similarly, K ∩ F (j0) = F since F (j0) is a subfield of F0.

Now consider the equation =j0(z) = 0. Any two roots ξ are related to each other

by some fractional linear transformation σ ∈ A5 defined over F (ζ5) ⊆ F0, so once we

adjoin one root we can generate all of them: K0 = F0(ξ). That is, Gal(K0/F0) is a

group of order not exceeding deg(=j0) = 60. K ·F0 ⊆ K0 because any root qν of q(x)

may be expressed as a rational function in ξ. (The roots of q(x) may be expressed in

terms of θν(ξ, 1) as described in the proof of 2.3.4.) The opposite inclusionK0 ⊆ K ·F0

is immediate from the inequality

[K0 : K · F0] =[K0 : F0]

[K · F0 : F0]=

[K0 : F0]

[K : K ∩ F0]=

[K0 : F0]

[K : F ]=

[K0 : F0]

60≤ 1 (2.30)

The equality K · F (j0) = K(j0) = F (θE,5) was proved in 2.3.4.


The reader may find it useful to keep the following diagram in mind as a summary

of the theorem:

K = F (q) 4 K(j0) = F (θE,5)2 K0 = K(ζ5, j0)

60

∣∣∣∣ 60

∣∣∣∣ 60

∣∣∣∣F 4 F (j0)

2 F0 = F (ζ5, j0)

(2.31)

To this point, we have considered how the splitting field K of a quintic over F is

related to the splitting field of certain principal quintics attached to elliptic curves.

In the next section, we will see that such splitting fields are also related to 5-division

points on elliptic curves.

2.5 5-Division Points on Elliptic Curves

Theorem 2.4.1 shows that the roots of the quintic may be expressed in terms of a

polynomial equation =j0(z) = 0 defined on an invariant j0. When we interpret this

quantity as the j-invariant of an elliptic curve, many properties will be revealed. We

describe this in detail in this section.

Let F be a field of characteristic relatively prime to 60, and let F a be its algebraic

closure. Define the rational function

j(z1, z2) =c4(z1, z2)

3

∆(z1, z2)= 1728

c4(z1, z2)3

c4(z1, z2)3 − c6(z1, z2)2(2.32)

Since this is a homogeneous rational function of degree 0, it defines a map j :

P1(F a)→ P1(F a) with the following zeroes and poles:

Inverse Image Points on the Zeroes of the Ramification

Icosahedron Polynomial Index

j−1(∞) 12 Vertices ∆(z1, z2) 5

j−1(0) 20 Midpoints of Faces c4(z1, z2) 3

j−1(1728) 30 Midpoints of Edges c6(z1, z2) 2

(2.33)


where infinity is considered as a point in the projective plane P1(F a). For any j0 ∈P1(F a) other than ∞, 0, or 1728, we can find ξ ∈ P1(F a) such that j(ξ, 1) = j0 by

means of solving the icosahedral equation =j0(ξ) = 0. This root ξ will be unramified

since the discriminant Disc(=j0) is nonzero; that is, we can find exactly 60 such points

in the preimage.

j(z1, z2) is invariant under any rotation in A5. Indeed, it is the quotient of two

invariant forms c4(z1, z2) and ∆(z1, z2). Hence j(z1, z2) is actually an isomorphism

from the quotient P1(F a)/A5 — in the sense of invariance under rotations — with

the projective line P1(F a) — in the sense of containing a point at infinity.

The field K0 = F (ζ5, z) of rational functions in z = z1/z2 is acted upon by the

fractional linear transformations S$, T$, and U$, but the rational function field

F0 = F (ζ5, j) is invariant under A5. Consider the following cubic curve E defined

over this field:

y2 = x3 − 3

122

j

j − 1728x− 2

123

j

j − 1728(2.34)

One checks that c4 = c4(z1, z2) γ−2 and c6 = c6(z1, z2) γ

−3 while ∆ = ∆(z1, z2) γ−6 is

nonzero, where γ = c6(z1, z2)/c4(z1, z2). This curve is indeed an elliptic curve, and

the j-invariant is j = j(z1, z2). In particular, these “coefficients” are homogeneous

rational functions of degree 0 which are invariant under A5, so they are well-defined

maps on the projective line P1(F a)/A5.

Using the formula for the 5-division polynomial in (1.22), we see that a point

P = (x, y) of order 5 has coordinates

x(z1, z2) =1

12

(z101 + 12 z8

1z22 − 12 z7

1z32 + 24 z6

1z42 + 30 z5

1z52

+ 60 z41z

62 + 36 z3

1z72 + 24 z2

1z82 + 12 z1z

92 + z10

2

)γ−1 (2.35)

y(z1, z2) =1

2

(z41z2 − 3 z3

1z22 + 4 z2

1z32 − 2 z1z

42 + z5

2

)·(z41z2 + 2 z3

1z22 + 4 z2

1z32 + 3 z1z

42 + z5

2

)2γ−3/2 (2.36)


(The irrationality of the y-coordinate need not concern us as we will only consider

the x-coordinates in the sequel.)

The 5-division polynomial ψ5 — when viewed as a homogeneous polynomial in z1

and z2 — has coefficients in the field F0, and so is invariant under the action of A5.

Hence, all of the roots of ψ5 are given by the functions(x◦σ

)(z1, z2) for σ ∈ A5. The

rational function field F (E[5]x) generated by these x-coordinates yields an icosahedral

extension.

Theorem 2.5.1 (A5 Extensions - Transcendental Form). Let F be a field of

characteristic relatively prime to 60, and E be an elliptic curve over the rational

function field F0 = F (ζ5, j) with j-invariant j = j(z1, z2).

Denote K0 = F (ζ5, z) as the field of all rational functions in z with coefficients

in F (ζ5), and F (E[5]x) as the rational function field generated by the x-coordinates

of the 5-division points as in (2.35). As transcendental extensions, K0 = F (E[5]x) is

generated by the x-coordinates, and Gal(K0/F0) ∼= A5.

Proof. Any elliptic curve E with j(E) = j(z1, z2) is in the form of equation (2.34)

after some change of variables, so we will assume that our curve is in this form.

(Recall that only irrational transformation involves the y-coordinate.)

First we prove that Gal(K0/F0) ∼= A5. The group A5 acts on K0 by fractional

linear transformations, so denote the function field fixed by such transformations by

F ′. Such a field has transcendence degree [K0 : F ′] = 60. However, F0 = F (ζ5, j) is

also a fixed field (since j = j(z1, z2) is invariant under action by A5) so F ′ ⊆ F0. j

has degree 60, so [K0 : F0] = 60 as well. This forces F ′ = F0 to be the field fixed by

A5, so Gal(K0/F0) ∼= A5 as asserted.

Next we show that K0 = F (E[5]x) is generated by the x-coordinates. (2.35) shows

that the rational function field F (E[5]x) is contained in K0 (i.e. the x-coordinates are

rational functions), so we have one inclusion. Conversely, let H = Gal(K0/F (E[5]x)

).

Since F (E[5]x) is the splitting field of the polynomial ψ5, H is a normal subgroup of

A5. The rational function in (2.35) shows that H has order not greater than 10, so

H must be trivial. Hence, K0 = F (E[5]x).

We conclude this chapter with a general result about icosahedral extensions. This


result will be used in the sequel when we consider the field fixed by the kernel of

icosahedral representations.

Corollary 2.5.2. Let F be a field of characteristic prime to 60, and let K be a normal

extension of F with Galois group A5. There exists a j0 in some quadratic extension

of F (√

5) such that for any elliptic curve E with j(E) = j0,

1. K(ζ5, j0) = F (E[5]x) is the field generated by the x-coordinates of the 5-torsion

of E;

2. K(j0) = F (ψ∗5) is the field generated by sum xP + x2P of x-coordinates of the

5-torsion.

F (E[5]x) is the splitting field of the 5-division polynomial ψ5(x), while F (ψ∗5) is

the splitting field of its resolvent ψ∗5(x) =∏

σ∈A5/D5

(x− [xσ(P ) + xσ (2P )]

)given by

ψ∗5(x) = x6 + b2 x5 + 10 b4 x

4

+ 40 b6 x3 + 80 b8 x2 + 16 (b2 b8 − b4 b6)x+ (−b2 b4 b6 + b22 b8 − 5 b26)

(2.37)

This resolvent was first studied by Annette Klute in [Klu97]. The motivation for such

a result came about from her research.

Proof. Choose a subgroup H ∼= A4 of index 5 in Gal(K/F ), and let KH be its fixed

field. Pick an element qν ∈ KH − F , and let q(x) be its minimal polynomial; then K

is its splitting field. We associate a j-invariant j0 defined over F (√

5,√

Tsch(q)) as

in Theorem 2.4.1. Let F0 = F (ζ5, j0), and E be any elliptic curve defined over F (j0)

such that j(E) = j0.

First we show that F (E[5]x) = K · F0. Theorem 2.4.1 states that K0 = K · F0

is the splitting field of the icosahedral polynomial =j0(z), so let ξ denote a root.

The x-coordinates of the 5-division points of E may be expressed in terms of the

polynomials x(ξ, 1) as in (2.35). This shows that the x-coordinates are contained

in K0; that is, F (E[5]x) ⊆ K0. Conversely, Theorem 2.5.1 states that F (E[5]x)


generates the splitting field of =j0(z), so we have inclusion the other way. Hence,

K(ζ5, j0) = K0 = F (E[5]x).

Now we show that F (ψ∗5) = K · F (j0). Theorem 2.4.1 also states that K(j0) =

K · F (j0) is the splitting field of the principal quintic θE,5, so it suffices to show that

F (ψ∗5) = F (θE,5). Since ψ∗5 is a degree 6 polynomial, we will show equality of fields

by introducing two more degree 6 polynomials.

Let θν be roots of θE,5, and consider the resolvent θ∗1 = 12√

5

∑ν (θν θν+1 − θν θν+2).

It is a root of the polynomial

θ∗E,5(x) = x6 + 5 c4 ∆x4 − 25 c24 ∆2 x2 − c4 c26 ∆2 x+ 5 (7 c26 + 8000 ∆) ∆3 (2.38)

It is well known that the splitting fields of a quintic and its resolvent sextic are equal;

that is, F (θE,5) = F (θ∗E,5). It suffices to show that F (ψ∗5) = F (θ∗E,5).

Now consider the rational function r∗(z1, z2) = 125 z61 z

62/

5√

∆(z1, z2). This form

is invariant under D5 = 〈S$, U$〉 so it satisfies a degree 6 polynomial. In fact, one

calculates that r∗ satisfies the equation (r2∗+10 r∗+5)3 = j0 r∗. The form also satisfies

the identities

xP + x2P = −b26− 1

6

c6c4

r2∗ + 10 r∗ + 5

r2∗ + 4 r∗ − 1

and θ∗1 =∆

c4(r∗ + 7) (r2

∗ + 10 r∗ + 5)

(2.39)

This shows that the splitting fields F (ψ∗5), F (θE,5), and F (θ∗E,5) are all equal to

splitting field of the equation (r2∗ + 10 r∗ + 5)3 = j0 r∗ and hence to each other.

We present another way to view the rational function r∗ described above, as

motivated in [Klu97]. Denote q = e2 πı τ for τ ∈ H in the upper half plane, and

X0(N) = H ∪ {∞} /Γ0(N) as the standard modular curve. When N = 1, we have

the canonical map j : X0(1)→ P1(C) defined by

j(τ) =1

q+ 744 + 196884 q + 21493760 q2 + . . . (2.40)


Now consider N = 5. We have a map r∗ : X0(5)→ X0(1) given by

r∗(τ) = 125

(η(5 τ)

η(τ)

)6

= 125 q ·∏n≥1

(1 + qn + q2n + q3n + q4n

)6(2.41)

where η(τ) = q1/24∏

n≥1 (1− qn) is the Dedekind eta function. By considering the

q-expansions, one checks that j = (r2∗ + 10 r∗ + 5)3/r∗ is a rational function in r∗. We

have another map z : X0(5)→ P1(C) which satisfies

z(τ)5 =η(5 τ)5

η(τ)= q ·

∏n≥1

(1 + qn + q2n + q3n + q4n

) (1− q5n

)4(2.42)

Again, by considering the q-expansions, one checks that r∗ = −125 z5/(z10+11 z5−1)

and hence j are rational functions in z. (We have already exhibited expressions for

c4, c6, and ∆ as polynomials in z.)

This may be summarized by the following diagram:

X0(5)z−−−→ P1(C)

r∗

y yj(z1,z2)

X0(1) −−−→j

P1(C)

(2.43)

The corollary above shows that it is not necessary to have ζ5 in the ground field of

definition, but in particular it does show that√

5 is necessary if one wishes to attach

the splitting field K to the field generated by the 5-division points on an elliptic curve.

In the next chapter, we will find that we can generalize these ideas to points

of order 3 and 4 as well. This will correspond to considering the rotations of the

tetrahedron and octahedron, respectively.

Chapter 3

The Platonic Solids

3.1 Introduction

We have already seen that rotations of the icosahedron are closely related to the

5-division points on elliptic curves. In this chapter, we generalize these results by

studying the rotations of the five Platonic solids: the tetrahedron, the cube, the

octahedron, the dodecahedron, and the icosahedron. We prove that the rotations

of the tetrahedron are associated with points of order 3 on elliptic curves, rotations

of both the cube and octahedron with points of order 4, and rotations of both the

dodecahedron and icosahedron with points of order 5.

In this chapter, N is either 3, 4, or 5 unless noted otherwise.

3.2 Geometric Realization of PSL2 (Z/N Z)

A Platonic solid is a regular convex polyhedron. Each face of the solid is a regular

polygon with the same number of vertices, and each vertex of the solid contains the

same number of edges. There are exactly five such polyhedra:

30

CHAPTER 3. THE PLATONIC SOLIDS 31

Platonic Solid No. of Faces No. of Vertices No. of Edges at each Vertex

Tetrahedron 4 triangular 4 vertices 3 edges

Cube 6 square 8 vertices 3 edges

Octahedron 8 triangular 6 vertices 4 edges

Dodecahedron 12 pentagonal 20 vertices 3 edges

Icosahedron 20 triangular 12 vertices 5 edges

(3.1)

The number of Platonic solids is limited by Euler’s formula, which states that the

number of faces minus the number of edges plus the number of vertices is a constant

(known as the Euler characteristic): F −E + V = 2. Both the cube with octahedron

and the dodecahedron with icosahedron are pairs of solids which are said to be dual

because the midpoints of the faces of one may be identified with the vertices of the

other. Equivalently, the number of faces of one is the number of vertices of the other,

a fact which can be verified directly by the information in the table above.

Proposition 3.2.1 (Rotations of the Regular Polyhedra). The group of sym-

metry for the tetrahedron is isomorphic to A4, the alternating group on four letters.

The groups of symmetry for both the cube and the octahedron are isomorphic to S4,

the full symmetric group on four letters. The groups of symmetry for both the do-

decahedron and the icosahedron are isomorphic to A5, the alternating group on five

letters.

Proof. The tetrahedron has four points, so its rotations are contained in S4. Ori-

entation must be preserved, so all of its rotations must be even; that is, the set of

rotations is A4.

The cube, on the other hand, consists of two copies of the tetrahedron — one

oriented 180◦ from the other. The set of rotations must be S4. If we identify rotations

about the vertices of the octahedron with rotations about the midpoints of the faces

of the cube, then we find that the rotations of the octahedron must also be S4.

Any rotation about the vertices of the dodecahedron must be order 3 (since each

vertex has 3 edges) so that the set of rotations is generated by 3 cycles. We may


identify such rotations about the vertices with rotations about the midpoints of the

faces of the icosahedron. There are 20 such faces, so we have a group of order 60.

Any such group with these properties must identically be A5.

From hence forth, we will identify rotations of the solids about the vertices with

rotations of their duals about the midpoints of the faces, so we consider only rotations

of the tetrahedron, the octahedron, and the icosahedron. The proposition above may

be summarized by saying that the group of rotations of these latter three solids is

PSL2 (Z/N Z) where N is the number of edges which meet at each vertex.

In order to write down explicit formulas for such rotations, we list explicit formulas

for the vertices solid as inscribed on the unit sphere. As in the last chapter, choose

the first point as P∞ = (1, 0, 0), and let S be the rotation that leaves this point fixed.

We know that N edges meet as this point (where N = 3, 4, or 5) so denote them as

Pν =

(cos θN , sin θN cos

2πν

N, sin θN sin

2πν

N

)for ν ∈ Z/N Z; P∞ = (1, 0, 0);

(3.2)

for some angle θN to be found. For the solids under consideration the faces are all

triangular so that the angle between P∞ and Pν is the same as that between Pν and

Pν+1. Then cos θ = P∞ ·Pν = Pν ·Pν+1 = cos2 θ+cos 2πN

sin2 θ, which we solve to find

cos θN =cos 2π

N

1− cos 2πN

=

−1

3N = 3;

0 N = 4;

1√5

N = 5.

sin θN =

2√

23

N = 3;

1 N = 4;

2√5

N = 5.

(3.3)

We have defined S as that rotation which maps Pν 7→ Pν+1. Another rotation is

a generator of order 2, denoted by T , that maps P∞ 7→ P0 and P1 7→ PN−1. If the

solid may be rotated 180◦ around the z-axis and brought back into itself — as is true

with each solid but the tetrahedron — then yet another rotation is the involution U

that maps Pν 7→ −P−ν . As matrices relative to the canonical basis in R3, these three

generators for PSL2 (Z/N Z) are given by


S =

1

cos 2πN− sin 2π

N

sin 2πN

cos 2πN

T =

cos θN sin θN

sin θN − cos θN

−1

U =

−1

−1

1

(3.4)

where θN is as given in (3.3).

It is easy to check the relations SN = T 2 = U2 = 1, U T = T U and USU−1 = S−1;

and that we have subgroups D2 = 〈T, U〉 (the Klein Viergruppe), ZN = 〈S〉 (the

cyclic group of order N), and DN = 〈S, U〉 (the dihedral group of order 2N). By

construction, our rotation groups are generated by S and T ; we leave it to the reader

to verify that U , whenever it exists as a rotation of the solid, may be expressed as a

combination of S and T .

As in the last chapter, stereographic projection $ gives a canonical isomorphism

with PSL2 (Z/N Z) and fractional linear transformations over C.

Proposition 3.2.1 (Projective PSL2 (Z/N Z) Representations). Fix N = 3,

4, or 5. Let {Pν} be the collection of vertices of the icosahedron as in (3.2), $ be

stereographic projection as in (2.3), and ς : {Pν} → P1(Z/N Z) be the surjective map

Pν 7→ (ν : 1).

Given an automorphism σ ∈ 〈S, T, U〉, the induced map σ$ = $ ◦ σ ◦ $−1 is a

fractional linear transformation in PGL2(C). Explicitly,

S$ z = ζN z, T$ z =$P0 z + 1

z −$P0

, U$ z = −1

z; (3.5)

where

ζN =

−1+

√−3

2N = 3;

√−1 N = 4;

−1+√

54

+

√10+2

√5

4i N = 5;

and $P0 =

1√2

N = 3;

1 N = 4;

1+√

52

N = 5.

(3.6)


Similarly, the induced map σς = ς ◦ σ ◦ ς−1 is a fractional linear transformation

in PSL2 (Z/N Z). Explicitly,

Sς (ν : µ) = (ν + 1 : µ), T ς (ν : µ) = (−µ : ν), U ς (ν : µ) = (z ν : z−1 µ).

(3.7)

where (Z/N Z)× = 〈z〉.In particular, PSL2 (Z/N Z) may be embedded in PGL2(C) as a projective 2-

dimensional representation, and 〈S, T, U〉 ∼= PSL2 (Z/N Z).

The reader may simplify the proposition above with the following diagram:

P1(C)$←−−− {Pν}

ς−−−→ P1(Z/N Z)

PGL2(C)

y 〈S,T,U〉y yPSL2(Z/N Z)

P1(C)$−1

−−−→ {Pν}ς−1

←−−− P1(Z/N Z)

(3.8)

Proof. Any automorphism σ ∈ 〈S, T, U〉 corresponds to an induced map σ$ =

$ ◦ σ ◦ $−1 as a fractional linear transformation. If we identify fractional linear

transformations as matrices in the projective linear group, then we have an explicit

representation of PSL2 (Z/N Z) as a subgroup of PGL2(C).

Now consider the projective line

P1(Z/N Z) =

{(ν : µ)

∣∣∣∣ ν ′ = λ ν, µ′ = λµ, gcd(λ, N) = gcd(ν, µ) = 1

}= (Z/N Z)× ∪

{(ν : µ)

∣∣∣∣ gcd(ν, µ) = 1, gcd(ν µ, N) > 1

} (3.9)

which has N∏

`|N(1+1/`) elements. By counting the vertices, we identify the tetra-

hedron with N = 3, the octahedron with N = 4, and the icosahedron with N = 5.

(By counting faces, we have similar identifications with the dual polyhedra.) Write

N as a power of ` in these cases, and consider the mapping ς from the vertices {Pν}to the the projective line given by


ς : (−1)ε P∞ 7→(1 : ε `

), (−1)ε Pν 7→

(ν + ε ` : 1

). (3.10)

whenever −Pν is a vertex on the Platonic solid. This map is surjective with trivial

kernel unless N = 5, in which case the kernel identifies antipodal points. (We include

this definition with ε to take care of the case when N = 4. For N = 3 the antipodal

point −Pν is not even a vertex.) Hence ς gives the desired bijection of the vertices

modulo equivalence with the projective line. We use this to find an equivalence with

the rotations of the Platonic solids and fractional linear transformations over Z/N Z.

GL2 (Z/N Z) acts on the projective line in the obvious way:(a b

c d

)·(ν : µ

)=(a ν + b µ : c ν + d µ

)(3.11)

We need a lemma to exhibit the generators of GL2 (Z/N Z).

Lemma 3.2.2. Given that N is the power of a prime, the general linear group

GL2 (Z/N Z) is generated by the three matrices

(1 1

1

),

(−1

1

), and

(a

d

)for a, d ∈ (Z/N Z)×. (3.12)

We digress briefly to prove the lemma. Consider the 2 × 2 matrix in (3.11), and

set D = a d − b c as the determinant. If c is not invertible modulo N = `n, then we

have the matrix product(a b

c d

)=

(−1

1

) (1 d

1

) (D + (1 + b) c (1 + b) d

−a −b

)(3.13)

where a is invertible. (If both c and a were divisible by `, then so would D.) Assume

without loss of generality that c is invertible. Then there is the decomposition(a b

c d

)=

(1 a/c

1

) (−1

1

) (c

D/c

) (1 d/c

1

)(3.14)


which proves the lemma.

We now show that the rotations of the Platonic solid are in one-to-one correspon-

dence with PSL2 (Z/N Z). Given any rotation σ ∈ 〈S, T, U〉 of the Platonic solid,

the induced map σς = ς◦σ◦ς−1 is a fractional linear transformation in PGL2 (Z/N Z).

Any such rotation must preserve the orientation of the Platonic solid, so the deter-

minant must be 1 modulo squares in (Z/N Z)×; that is, σς ∈ PSL2 (Z/N Z). But

the maps listed in the proposition above are precisely the projective images of the

generators listed in the lemma, so the map ϕ 7→ ϕσ is the desired isomorphism.

3.3 Invariant Forms and Principal Polynomials

We attach invariant forms to each Platonic solid as we did with the icosahedron. This

section is motivated by Klein [Kle93]. We assume that all calculations are done over

a field F of characteristic relatively prime to the order of the set of rotations.

Consider the following invariant form which vanishes at all of the vertices of the

given solid:

f(z1, z2) = z2

∏σ

[ z1 −$ (σ P0) z2 ] over all σ ∈ 〈S, T, U〉. (3.15)

If two forms f1(z1, z2) and f2(z1, z2) are left invariant under a set of rotations,

then so Covariant(f1, f2) = det(

∂fi

∂zj

). We use this idea to define the three invariant

forms

∆ = f(z1, z2)N , c4 = Covariant

(∂f

∂z1

,∂f

∂z2

), c6 = Covariant (f, c4) ; (3.16)

which will be suitably normalized so that c34 − c26 = 1728 ∆. We choose the following

homogeneous polynomials:


N = 3 :

∆(z1, z2) =(

32z2 (2√

2 z31 − z3

2))3

c4(z1, z2) = 18 z1 (z31 + 2

√2 z3

2)

c6(z1, z2) = 54√

2 (z61 − 5

√2 z3

1 z32 − z6

2)

(3.17)

N = 4 :

∆(z1, z2) =(−16 z1 z2 (z4

1 − z42))4

c4(z1, z2) = 16 (z81 + 14 z4

1 z42 + z8

2)

c6(z1, z2) = 64 (z121 − 33 z8

1 z42 − 33 z4

1 z82 + z12

2 )

(3.18)

N = 5 :

∆(z1, z2) =(−z1 z2 (z10

1 + 11 z51 z

52 − z10

2 ))5

c4(z1, z2) =(z201 + z20

2

)− 228

(z151 z5

2 − z51 z

152

)+ 494 z10

1 z102

c6(z1, z2) =(z301 + z30

2

)+ 522

(z251 z5

2 − z51 z

252

)− 10005

(z201 z10

2 + z101 z20

2

)(3.19)

∆(z1, z2) vanishes on the vertices, while c4(z1, z2) vanishes on the midpoints of the

faces and c6(z1, z2) vanishes on the midpoints of the edges. Indeed, if the characteristic

of the field F is prime to the order of PSL2 (Z/N Z) then these points are distinct;

consider the following discriminants as proof:

N PSL2 (Z/N Z) Order Disc(∆1/N) Disc(c4) Disc(c6)

3 A4 12 −1237 −212 315 218 345

4 S4 24 412 296 34 2200 324

5 S5 60 525 260 310 595 −290 360 5205

(3.20)

When the set of rotations of the Platonic solid has order |PSL2 (Z/N Z) | = 12n,

then ∆(z1, z2), c4(z1, z2) and c6(z1, z2) are homogeneous of degree 12n, 4n and 6n,

respectively. It is also easy to see that both c34 and ∆ are homogeneous polynomials

in zN1 and zN

2 .


The symmetry group PSL2 (Z/N Z) has an N cycle, so we expect to associate a

polynomial of degree N to these forms. In order to do so, we exhibit a homogeneous

rational function r(z1, z2) which is invariant under 〈T$, U$〉 but acted upon nontriv-

ially by S$. We use the idea of chapter 2, where we exhibited a rational function

which vanishes at the fixed points of T$, U$, and T$ U$, to arrive at the following

functions:

N r(z1, z2)

336 [z2

1 − 2 ($P0) z1 z2 − z22 ]

2

∆(z1, z2)1/3

48 [z2

1 − (1 + i) z1 z2 − i z22 ] [z2

1 − (1− i) z1 z2 + i z22 ] [z2

1 + (1 + i) z1 z2 − i z22 ]

∆(z1, z2)1/4

5[z2

1 + z22 ]

2[z2

1 − 2 ($P0) z1 z2 − z22 ]

2[z2

1 − 2 ($P0)−1 z1 z2 − z2

2 ]2

∆(z1, z2)1/5

(3.21)

The following lemma is a generalization of lemma 2.3.1. It shows how to attach a

polynomial of degree N to these forms.

Lemma 3.3.1 (Klein). Fix N as 3, 4, or 5. Let F be a field of characteristic not

dividing |PSL2 (Z/N Z) |, and r = r(z1, z2) be the rational function defined in (3.21).

For arbitrary constants m and n in the algebraic closure F a, there are homogeneous

forms θN, ν(z1, z2) such that the polynomial θN(x) =∏

ν(x− θN, ν) is principal (i.e. it

has no xN−1 or xN−2 terms) and its coefficients are rational functions of m, n, and

j = c34/∆. Explicitly,


N 3 4 5

θN, ν1

rν + 12

m

rν

m

rν + 3+

n

(rν + 3) (r2ν + 10 rν + 45)

θN(x) x3 − 1

jx4 − (1 + i)7m3

jx− m4

jx5 + Ax2 +B x+ C

j (r + 12)3 r3(r + (1 + i)7

)(r + 3)3 (r2 + 11 r + 64)

(3.22)

where rν = r ◦ (S$)ν and the coefficients A, B, and C are as in Lemma 2.3.1.

Proof. This can be verified directly using a symbolic calculator. (The reader may also

consult [Kle93], although Klein assumes one is working over C. The form r = r(z1, z2)

used for N = 4 is slightly different as well.)

Every principal polynomial of degree less than 6 can be put into the form of one

of the polynomials above.

Lemma 3.3.2 (Klein). Fix N as 3, 4, or 5. Let F be a field of characteristic not

dividing |PSL2 (Z/N Z) |, and let θ(x) be a principal polynomial of degree N defined

over F . Then there exist m, n, and j in F if N = 3, F (√−1) if N = 4 and

F (√

5Disc(θ)) if N = 5 such that the roots of θ(x) are in the form θN, ν as defined

in Lemma 3.3.1.

Proof. When N = 3, all principal cubics are in the form θ(x) = x3 + A, so choose

j = −1/A. When N = 4, all principal quartics are in the form θ(x) = x4 +Ax+B so

choose m = (1 + i)7B/A and j = 47B3/A4. The case where N = 5 is a restatement

of lemma 2.3.2.

The Tschirnhaus Transformation used in the previous chapter works for any poly-

nomial, but we will be concerned with the result for polynomials of degree less than

6.


Lemma 3.3.3 (Tschirnhaus). Through a quadratic substitution, every polynomial

of degree less than 6 can be transformed into a principal one.

That is, fix N as 3, 4, or 5, let F be a field of characteristic not dividing the order

of PSL2 (Z/N Z), and denote q(x) as a polynomial of degree N defined over F with

roots qν. Then there exist α, β, and γ in some quadratic extension F(√

Tsch(q))

such that the polynomial polynomial defined over this quadratic extension, with roots

α q2ν + β qν + γ, has its xN−1 and xN−2 terms missing.

Proof. Say that q(x) =∑

µ aµ xµ = aN

∏ν(x−qν), and form the resolvent polynomial

θ(x) =∏

ν(x− θν) with roots θν = α q2ν + β qν + γ for some α, β, and γ to be found.

Let N = 3. To set the x2 and x terms to zero, we define α, β, and γ as

α = 3 a3

(a2

2 − 3 a1 a3

)β =

3 (2 a32 − 7 a1 a2 a3 + 9 a0 a

23) + 3 a3

√Tsch(q)

2

γ =3 (a1 a

22 − 4 a2

1 a3 + 3 a0 a2 a3) + a2

√Tsch(q)

2

(3.23)

where the Tschirnhaus discriminant is

Tsch(q) = −3 Disc(q) = −3(a2

1 a22 − 4 a0 a

32 − 4 a3

1 a3 + 18 a0 a1 a2 a3 − 27 a20 a

23

)(3.24)

Hence θ(x) = x3 + A has coefficients in in the quadratic extension F (√

Tsch(q)).

Now let N = 4. To set the x3 and x2 terms to zero, we define

α = 2 a4

(3 a2

3 − 8 a2 a4

)β = 2

(3 a3

3 − 10 a2 a3 a4 + 12 a1 a24

)+ 4 a4

√Tsch(q)

γ = 2(a2 a

23 − 4 a2

2 a4 + 3 a1 a3 a4

)+ a3

√Tsch(q)

(3.25)

where the Tschirnhaus discriminant is


Tsch(q) = 2 (−a22 a

23 + 4 a3

2 a4

+ 3 a1 a33 − 14 a1 a2 a3 a4 + 18 a2

1 a24 + 6 a0 a

23 a4 − 16 a0 a2 a

24)

(3.26)

Hence θ(x) = x4 +Ax+B has coefficients in the quadratic extension F (√

Tsch(q)).

The case where N = 5 is a restatement of lemma 2.3.3, so the proof is complete.

Putting these three lemmas together, we arrive at the key result of this section.

It is the corresponding generalization of Theorem 2.3.4.

Theorem 3.3.4. Fix N as 3, 4, or 5. Let F be a field of characteristic not dividing

|PSL2 (Z/N Z) |, q(x) be a polynomial of degree N defined over F , and denote K as

its splitting field. Denote Tsch(q) as given in (3.24) for N = 3, (3.26) for N = 4,

and (2.20) for N = 5. Moreover, assume that√−1 ∈ F if N = 4.

There exists j0 in a quadratic extension of F(√

Tsch(q))

such that for any elliptic

curve E in Weierstrass form with j(E) = j0, K(j0) is the field generated by the roots

of either jN(r) = j0 or θE,N(x) = 0, where

jN(r) =

(r + 12)3

r3(r + (1 + i)7

)(r + 3)3 (r2 + 11 r + 64)

θE,N(x) =

x3 −∆

x4 + 32 ∆x+ 4 c4 ∆

x5 − 40 ∆x2 − 5 c4 ∆x− c24 ∆

(3.27)

The polynomials θE,3(x) and θE,4(x) are those found in the proof Proposition 1.3.1

and (1.20), respectively. This theorem explains their importance in more detail. Note

that√−1 ∈ F when N = 4 so that the polynomial j4(r) = r3

(r+(1+ i)7

)is defined

over F .

Proof. Let q(x) =∏

ν(x− qν), and choose α, β, and γ as in the proof of lemma 3.3.3

so that θν = α q2ν + β qν + γ are the roots of a principal polynomial θ(x). Following


lemma 3.3.2, choose m, n, and j0 such that α q2ν +β qν +γ = θN,ν is a rational function

of rν = r ◦ (S$)ν as described in lemma 3.3.1. Hence the roots of q(x) are in one-to-

one correspondence with the roots of jN(r) = j0. In particular, j0 is an element of

F(√

Tsch(q))

if N = 3 or N = 4, and an element of F(√

5 Disc(q),√

Tsch(q))

if

N = 5.

The equation jN(r) = j0 does not depend on m and n — as long as m and n are

constants in F (j0). Given any elliptic curve E with j(E) = j0, form the principal

polynomials

θE,3(x) =∏ν

(x− c4

rν+12

)θE,4(x) =

∏ν

(x− (1−i) c4

rν

)θE,5(x) =

∏ν

(x− c4

rν+3

)(3.28)

The roots of θE,N(x) are in one-to-one correspondence with the roots of jN(r) = j0,

so the proposition follows.

3.4 N-Division Points on Elliptic Curves

Given a polynomial q(x) defined over F with degree less than 6, there is an associated

quantity j0 defined over the algebraic closure F a. Conversely, given j0 in F a, we can

solve the equation j0 = c34/∆ = j(z1, z2) for ξ = z1/z2, where the j-invariant is the

rational function

j(z1, z2) =c4(z1, z2)

3

∆(z1, z2)= 1728

c4(z1, z2)3

c4(z1, z2)3 − c6(z1, z2)2(3.29)

j(z1, z2) is a homogeneous rational function of degree 0, so it defines a map

P1(F a) → P1(F a) from the projective line to itself with the following zeroes and

poles:


Inverse Image Points on the Zeroes of the Ramification

Platonic Solid Polynomial Index

j−1(∞) Vertices ∆(z1, z2) N

j−1(0) Midpoints of Faces c4(z1, z2) 3

j−1(1728) Midpoints of Edges c6(z1, z2) 2

(3.30)

For any j0 ∈ F a other than 0, 1728, or ∞, we can find ξ ∈ F a that is a root of the

polynomial =j0,N(z) = c4(z, 1)3 − j0 ∆(z, 1). We define the tetrahedral, octahedral,

and icosahedral equations as =j0,N(z) = 0 for N = 3, N = 4, and N = 5, respectively,

where

=j0,3(z) = 1728 z3[z3 + 2

√2]3

+ j0

[1− 2

√2 z3]3

=j0,4(z) = 256[z8 + 14 z4 + 1

]3 − j0 z4[z4 − 1

]4=j0,5(z) =

[(z20 + 1

)− 228

(z15 − z5

)+ 494 z10

]3

+ j0

[z5(z10 + 11 z5 − 1

)5](3.31)

We claim that these equations have distinct roots if the characteristic of F does

not divide N . Indeed, the discriminants of the polynomials are

N 3 4 5

Disc(=j0,N) −3147 j80 (j0 − 1728)6 4332 j16

0 (j0 − 1728)12 5785 j400 (j0 − 1728)30

(3.32)

Once we solve this equation for a root ξ, we find r(ξ, 1) and then θN,ν(ξ, 1) by lemma

3.3.1. The roots of q(x) may be expressed in terms of θN,ν(ξ, 1) as described in

the proof of Theorem 3.3.4. The following theorem states the explicit relationship

between the splitting fields of q(x), θE,N(x), and =j0,N(z). It is the corresponding

generalization of Theorems 2.4.1 and 2.5.1.

Theorem 3.4.1 (PSL2 (Z/N Z) Extensions). Fix N as 2, 3, 4, or 5, let F be a field


of characteristic not dividing the order of PSL2 (Z/N Z), and let K/F be a normal

extension with Galois group PSL2 (Z/N Z). Moreover, assume that√−1 ∈ F when

N = 4.

There exists j0 in a biquadratic extension of F such that for any elliptic curve E

with j(E) = j0, K is contained in the splitting field of ψN(x). More precisely, set

F0 = F (j0, ζN) and denote K0 = F (E[N ]x) as the field generated by the x-coordinates

of the N-division points. Then K0 = K · F0 and we have the isomorphisms

Gal(K/F ) ∼= Gal(K(j0)/F (j0) ) ∼= Gal(K0/F0 ) ∼= PSL2 (Z/N Z) (3.33)

During the proof of Theorem 3.3.4, we found that the j0 is contained in the

quadratic extension F(√

Tsch(q))

if N is 3 or 4, so in particular we may be able to

find the elliptic curve E as in the theorem above which can be defined over F (√−3)

if N = 3, F (√−1) if N = 4, or F (

√5) if N = 5. These are the smallest field of

definitions for the elliptic curves.

Proof. The statement above withN = 2 is corollary 1.2.2, and withN = 5 is Theorem

2.4.1, so we consider only the cases where N = 3 and N = 4.

In these cases A4 ⊆ Gal(K/F ) ⊆ S4, so let q(x) be a quartic polynomial defined

over F such that K is its splitting field. We must show that we can associate a j0

in some biquadratic extension of F such that K · F0 = K0 and K ∩ F0 = F where

F0 = F (j0, ζN) and K0 is the field generated by the x-coordinates of the N -division

points. Once we do, then K ∩ F (j0) = F as well since F (j0) ⊆ F0, and

PSL2 (Z/N Z) ∼= Gal(K/F )

∼= Gal(K/K ∩ F (j0)

) ∼= Gal(K · F (j0)/F (j0)

) ∼= Gal(K(j0)/F (j0)

)∼= Gal

(K/K ∩ F0

) ∼= Gal(K · F0/F0

) ∼= Gal(K0/F0

) (3.34)

We exploit the fact that every quartic has a resolvent cubic; we use this to distin-

guish between the cases N = 3 and N = 4. Set q(x) =∑

ν aν xν = a4

∏4ν=1(x− qν),


denote q∗1 = 13a2−a4 (q1 q2+q3 q4), and consider the polynomial q∗(x) =

∏3ν=1(x−q∗ν).

It has the expansion

q∗(x) = x3 − a22 − 3 a1a3 + 12 a0a4

3x+

2 a32 − 9 a1a2a3 + 27a0a

23 + 27a2

1a4 − 72a0a2a4

27(3.35)

It is easy to check Disc(q) = Disc(q∗). Consider the two polynomials

θ(x) =4∏

ν=1

[x−

(α q2

ν + β qν + γ)]

and θ∗(x) =3∏

ν=1

[x−

(α q∗ν

2 + β q∗ν + γ)](3.36)

for some α, β, and γ to be found. We now distinguish between the two cases:

Case 1: N = 3. The strategy of the proof in this case is to explain the diagram

K 2 F (ψ3)= F (E[3]x)

4

∣∣∣∣ 4

∣∣∣∣ 4

∣∣∣∣F (q∗) 2 F (θE,3)

= F0(3√

∆)

3

∣∣∣∣ 3

∣∣∣∣ 3

∣∣∣∣F 2 F (j0)

= F0

(3.37)

Choose α, β, and γ such that θ∗(x) is a principal cubic. The proof of lemma

3.3.3 shows that the resulting polynomial has coefficients in F (√

Tsch(q∗)). Now

Tsch(q∗) = −3 Disc(q∗) = −3 Disc(q), and by assumption Gal(K/F ) ∼= A4 so Disc(q)

is a square. Hence, θ∗(x) = x3 + A has coefficients in F (√−3), so choose j0 = −1/A

in F (√−3) and let E be any elliptic curve with j(E) = j0.

Let F (q∗) be the splitting field of the resolvent cubic. By Theorem 3.3.4, we have

F (q∗, j0) = F (θE,3). Proposition 1.3.1 shows that θE,3(x) is the resolvent cubic of

the polynomial ψ3(x), so by comparing resolvent cubics we have K(j0) = F (q, j0) =

F (ψ3). Clearly, F (j0) = F (ζ3) = F0. This shows that F (E[3]x) = K(j0) = K · F0.

We now show that K ∩F0 = F . The extension F (√−3)/F has degree dividing 2,

so K ∩ F0/F also has degree dividing 2. However, A4 has no subgroups of index 2,


so we must have K ∩ F0 = F .

Case 2: N = 4. The strategy in this case is to explain the diagram

K 2 F (θE,4)= F (E[4]x)

4

∣∣∣∣ 4

∣∣∣∣ 4

∣∣∣∣F (q∗) 2 F (θ∗E,4)

= F (E[2]x)

6

∣∣∣∣ 6

∣∣∣∣ 6

∣∣∣∣F 2 F (j0)

= F0

(3.38)

Recall that it is assumed that√−1 ∈ F .

Choose α, β, and γ such that θ(x) is a principal quartic. The proof of lemma

3.3.3 shows that the resulting polynomial has coefficients in F (√

Tsch(q)). Hence,

θ(x) = x4 + Ax + B, so choose j0 = 47B3/A4 and let E be any elliptic curve with

j(E) = j0.

By Theorem 3.3.4, K(j0) = F (θE,4). Proposition 1.4.2 shows that the splitting

field of θE,4(x) is the splitting field of ψ4(x), so K · F (j0) = F (E[4]x) = K0. (By

comparing resolvent cubics, we also have F0(q∗) = F (θ∗E,4) where θ∗E,4(x) is defined in

(1.20). One also checks by explicit manipulation of the roots of θ∗E,4(x) that F (θ∗E,4) =

F (E[2]x).)

It remains to show thatK∩F0 = F . The extension F0/F is quadratic, soK∩F0/F

has at worst degree 2. Hence√

Tsch(q) ∈ F (√

Disc(q)). We must show that in the

expression√

Tsch(q) = α+ β√

Disc(q) for some α and β in F , we must have β = 0.

Upon squaring this expression, we find αβ = 0 since√

Disc(q) is irrational, so assume

that α = 0. We will find a contradiction.

Recall that Disc(q) = Disc(q∗) = 4A30 − 27B2

0 in terms of the resolvent cubic

q∗(x) = x3 − A0 x + B0 as in (3.35). The Tschirnhaus discriminant in (3.26) may

be expressed as Tsch(q) = 36 a4B0 − 2 (3 a23 − 8 a2a4)A0. Since Tsch(q) = β2 Disc(q)

we must be able to express Tsch(q) as a rational function of A0 and B0 alone, which

means 3 a23−8 a2a4 can be expressed in terms of A0 and B0. The roots of q∗(x) are in

the form q∗1 = 13a2−a4 (q1 q2 + q3 q4), so the only expression possible is 3 a2

3− 8 a2a4 =

−8 a4

∑ν q

∗ν = 0, in which case


Tsch(q) =

(48 a1 a

24 − 3 a3

3

8 a4

)2

=⇒√

Disc(q) = ± 1

β

48 a1 a24 − 3 a3

3

8 a4

∈ F (3.39)

which is indeed a contradiction. Hence, K ∩ F0 = F , which completes the proof of

the theorem.

We conclude this chapter by noting that the elliptic curve

y2 = x3 − 3

122

j

j − 1728x− 2

123

j

j − 1728(3.40)

with j-invariant j = j(z1, z2) is defined over rational function field F0 = F (j, ζN)

and hence is invariant under any rotation of the Platonic solid. The x-coordinates of

the N -torsion points are in the form(x ◦ σ

)(z1, z2) for rotations σ ∈ PSL2 (Z/N Z),

where

x(z1, z2) =1

12 γ·

9√

2 z21 (N = 3);

4 (5 z41 − z4

2) (N = 4);(z101 + 12 z8

1 z22 − 12 z7

1 z32 + 24 z6

1 z42 + 30 z5

1 z52

+60 z41 z

62 + 36 z3

1 z72 + 24 z2

1 z82 + 12 z1 z

92 + z10

2

) (N = 5).

(3.41)

with corresponding y-coordinate

y(z1, z2) =1

4 γ3/2·

4√

2(2√

2 z31 − z3

2

)(N = 3);

8 z21 (z4

1 − z42) (N = 4);

2(z41 z2 − 3 z3

1 z22 + 4 z2

1 z32 − 2 z1 z

42 + z5

2

)·(z41 z2 + 2 z3

1 z22 + 4 z2

1 z32 + 3 z1 z

42 + z5

2

)2 (N = 5).

(3.42)

given that γ = c6(z1, z2)/c4(z1, z2).


In the remaining chapters, we discuss the L-series attached to elliptic curves as-

sociated with prescribed PSL2 (Z/N Z)-extensions.

Chapter 4

Representations and L-Series

4.1 Introduction

In this and the remaining chapters, we study the properties of a specific type of

L-series: those that arise from Galois representations over number fields.

We invoke the following standard definitions. Let F be a number field i.e. a finite

extension of the field of rational numbers Q. Given the algebraic closure F a, we define

the profinite group GF to be the absolute Galois group:

GF = Gal(F a/F ) = lim←−Gal(L/F ) over all finite L/F . (4.1)

Fix a positive integer n and let V be an n-dimensional vector space over a field of

characteristic 0. The collection of invertible transformations GL(V ) is also a group,

so consider continuous group homomorphisms ρ : GF → GL(V ). We say that ρ is a

complex Galois representation if V ∼= Cn, and, for a fixed prime `, that ρ is an `-adic

Galois representation if V ∼= Qn` .

Recall from Galois Theory that if L is the fixed field of a closed normal subgroup

H in GF , then H = GL, and we have the isomorphism Gal(L/F ) ∼= GF/GL. In

particular, if H = ker(ρ) for some continuous representation ρ, then H is such a

subgroup of GF so that its fixed field L is Galois over F . In this case GL = ker(ρ)

and Gal(L/F ) ∼= im(ρ). (We will work with complex representations which factor

49

CHAPTER 4. REPRESENTATIONS AND L-SERIES 50

through a finite quotient of GF , whereas the `-adic representations we will choose

will not have finite image.) Any representation ρ has a projectivization ρ defined by

composition with the map GL(V ) → PGL(V ). If K is the field fixed by the kernel

of such a composition, then GK = ker(ρ) and Gal(K/F ) ∼= im(ρ).

The reader should keep the following diagram in mind:

1 −−−→ Gal(L/K) −−−→ Gal(L/F ) −−−→ Gal(K/F ) −−−→ 1y yρ

yρ

1 −−−→ Z −−−→ GL(V ) −−−→ PGL(V ) −−−→ 1

(4.2)

where Z is the center of GL(V ).

4.2 The Frobenius Element

In order to better understand Galois representations in general, we must consider the

underlying structure of the ramification groups. This section is included to be more

of a review for the reader, and a reference for the definition of notation for the sequel.

The results stated below are contained in Serre [Ser79].

Let F be a number field, and L be a Galois extension of F . A prime p in F

splits into a product p = Pe1 Pe

2 · · · Per of r = r(P|p) primes P in L, each with

ramification index e = e(P|p). Denote OF and OL as the ring of integers of F and

L, respectively. For each P, the quotients F = OF/p and L = OL/P are both finite

fields. The degree of the extension is f = f(P|p); if F has N p = q elements, then L

has N P = qf elements. If L/F is finite, then [L : F ] = r f e.

For each prime P above p, the subgroup DP in Gal(L/F ) acts trivially on P.

Equivalently, the decomposition group is defined as the subgroup which acts on the

finite field L = OL/P; it has index r, the number of prime ideals P lying above

p. Given an automorphism τ ∈ DP and a coset xP ∈ L, the map xP 7→ (τx) P

induces a map DP → Gal(L/F ). The kernel is the inertia group TP; it has order e,

the ramification index.

The quotient DP/TP is isomorphic to the cyclic Galois group Gal(L/F ) of order

f . This group is the same — up to conjugation by σ ∈ Gal(L/F ) — for all primes P


since DσP/TσP = σ DP/TP σ−1. The generator of this group is the coset σP = σP TP,

the so-called Frobenius element defined by the congruence

σP(x) ≡ xNp (mod P) for all x ∈ OL. (4.3)

One generalizes the construction above by way of the higher ramification groups

Gi. For each nonnegative integer i, let Gi be the subgroup of Gal(L/F ) which acts

trivially on the ring OL/Pi+1. Then G0 = TP is the inertia group, and we define

G−1 = DP as the decomposition group. In general, Gi+1 is a normal subgroup of Gi,

the quotient G0/G1 is cyclic, and if L has characteristic p then the quotient Gi/Gi+1

is an elementary abelian p-group for all positive integral i.

4.3 L-Series Attached to Galois Representations

This section is an adaptation of Lang [Lan94] and Klute [Klu97]. We continue to

assume that F is a number field.

Consider the case where V ∼= C is a 1-dimensional complex vector space. Given

a representation χ : GF → C×, we associate a Hecke character on ideals n ⊆ OF by

defining χ(n) =∏

p χ(σP)f(n,p) when n =∏

p pf(n,p), and χ(p) = 0 when p is ramified.

(We do so because the Frobenius element is not uniquely defined in this latter case.)

We define the L-series as

L(χ, s) =∏

p

(1− χ(p)

N ps

)−1

(4.4)

which is reminiscent of the L-series attached to Dirichlet characters. Hecke [Hec87]

proved that as long as χ 6= χ0 is not the trivial character (i.e. for all σ ∈ GF we

define χ0(σ) = 1 ) then the corresponding L-series is an entire function.

Now consider the more general scenario where ρ : GF → GL(V ) as a finite dimen-

sional representation — which may be either complex or `-adic depending on when

V ∼= Cn or V ∼= Qn` . For each nonnegative integer m, define the endomorphism


ρ(pm) =1

|σP|∑σ∈σP

ρ (σm) for each p in OF . (4.5)

This map is independent of the choice of prime P|p — up to conjugation by some

ρ(σ). (One may think of this quantity sum as the “average” over the coset (σP)m.)

Upon expressing the coset as σP = σP TP, one finds the factorization

ρ(pm) = ρ(σP)m ερ(p) where ερ(p) =1

|TP|∑τ∈TP

ρ(τ). (4.6)

The endomorphism ερ(p) is an idempotent in End(V ); that is, ερ(p) · ερ(p) = ερ(p)

for any prime p in OF .

The endomorphism ρ(pm) depends on a choice of prime P lying above, but the

trace tr ρ(p) and determinant det ρ(p) do not. For a variable T , we define

Lp(ρ, T ) = exp

[−

∞∑m=1

tr ρ(pm)Tm

m

](4.7)

which is both independent of the choice of prime P and the choice of a basis for V .

We define the L-series associated to ρ as the product

L(ρ, s) =∏

p

Lp(ρ, N p−s)−1 =∏

p

det[I − ρ(p) N p−s

]−1(4.8)

over all nonzero primes p. There are other ways to exhibit this product.

Proposition 4.3.1. Let ρ : GF → GL(V ) be a Galois representation, where V is

either a complex or `-adic vector space.

1. The Euler factor corresponding to the prime p in F is also equal to

Lp(ρ, T ) = det [I − ρ(p)T ] = det

[I − ρ(σP)

∣∣∣∣V Tp

T

](4.9)

where ρ|V Tp is that linear transformation restricted to the invariant subspace

V Tp ⊂ V .


2. The L-series associated to ρ may also be expressed in the form

L(ρ, s) =∑

n

a(n)

N nswhere a(n) =

∏p|n

a(pf(n,p)

)(4.10)

as the sum over all nonzero ideals n ⊆ OF in the ring of integers of F . Moreover,

the a(n) are in the field of definition for V .

3. a(p) = tr ρ(p) is the trace of Frobenius.

Proof. First we prove the statement about the Euler factors. Say that V is an n-

dimensional vector space so that the endomorphism ρ(p) has n eigenvalues λν . Then

we compute

exp

[−

∞∑m=1

tr ρ(pm)Tm

m

]= exp

[−

n∑ν=1

(∞∑

m=1

λmν

Tm

m

)]=

n∏ν=1

exp [log (1− λν T )]

(4.11)

which is equal to det [I − ρ(p)T ].

Now consider the idempotent ερ(p). If the representation ρ(τ) = I is the identity

for all τ ∈ TP in the inertia group, then ερ is the identity matrix as well. In general, the

collection of matrices ρ(τ) has an invariant subspace V Tp ⊂ V , where the dimension

of this subspace is equal to the rank of ερ(p). Then

Lp(ρ, T ) = det

[I − ρ(σP) ερ(σP)T

]= det

[I − ρ(σP)

∣∣∣∣V Tp

T

](4.12)

Second we prove the statements about the infinite sum. Again, assume that ρ(p)

has the n eigenvalues λν , and consider det [I − ρ(p)T ] as a polynomial in T . (This

polynomial does not have degree n in general; it has degree equal to the rank of the

idempotent ερ(p).) It has the expansion

det [I − ρ(p)T ] =∑

µ

λ(µ) T µ where λ(µ) = (−1)µ∑

ν1<···<νµ

λν1λν2 . . . λνµ

(4.13)


The reciprocal of this polynomial is det [I − ρ(p)T ]−1 =∑

f a(pf )T f , where the a(pf )

are defined by the recursion relation

λ(0) a(pf+n) + λ(1) a(pf+n−1) + λ(2) a(pf+n−2) + · · ·+ λ(n) a(pf ) = 0 (4.14)

A closed form solution to this recursion is given by

a(pf ) =∑

f1+···+fn=f

λf1

1 λf2

2 . . . λfnn (4.15)

These coefficients are in the same field over which the vector space V is defined since

they are symmetric in the eigenvalues. In particular, a(p) = tr ρ(p) is just the sum of

the eigenvalues.

To finish the proof, write

L(ρ, s) =∏

p

Lp(ρ, N p−s)−1 =∏

p

(∑f

a(pf ) N p−s f

)(4.16)

and compare coefficients.

The invariant subspace V Tp in the proposition above is part of a filtration of

subspaces. Let L be the fixed field of ker(ρ), and consider the Galois group Gal(L/F ).

Recall that the inertia group G0 = Tp contains subgroups Gi, the higher ramification

groups. Define the conductor fρ as the ideal

fρ =∏

p

pf(ρ,p) where f(ρ, p) =∞∑i=0

1

(G0 : Gi)codimV Gi . (4.17)

This sum is actually finite since V is assumed to be finite dimensional. The exponent

f(ρ, p) consists of two parts: The tame part of the conductor is ε(ρ, p) = codimV G0 ,

whereas the wild part of the conductor δ(ρ, p) comes about only if the prime p is

wildly ramified; i.e. the higher ramification groups are nontrivial. Note that the

exponent f(ρ, p) is zero if the prime p is not ramified.


f(ρ, p) = ε(ρ, p) + δ(ρ, p) where

ε(ρ, p) = codimV G0 = nullity of ερ(p)

δ(ρ, p) =∞∑i=1

1

(G0 : Gi)codimV Gi

(4.18)

Any finite dimensional representation ρ : GF → GL(V ) decomposes into smaller

representations if V decomposes into invariant subspaces. That is, if V =⊕

ν Vν

then ρ =⊕

ν ρν where ρν = ρ|Vν is just the restriction to the corresponding subspace.

From (4.7), we find

Lp

(⊕ν

ρν , T

)= exp

[−

∞∑m=1

tr

(⊕ν

ρν(pm)

)Tm

m

]=∏ν

Lp(ρν , T ) (4.19)

so that the L-series factors as

L(ρ, s) =∏ν

L(ρν , s) whenever ρ =⊕

ν

ρν . (4.20)

Recall that a representation ρ is irreducible if V has no invariant subspaces. By

definition, 1-dimensional representations are irreducible. From hence forth we study

the properties of irreducible 2-dimensional representations.

4.4 `-adic Representations from Elliptic Curves

In this section we work with `-adic representations coming from elliptic curves; that

is, we set V ∼= Q2` . To this end, we consider the representation based on the N -

division points of an elliptic curve for N = `n, then proceed to show that the L-series

corresponding to the `-adic representation is indeed the usual definition for the L-

series of an elliptic curve. Most of the material found in this section is standard, but

we include the discussion to fix the notation. We set N = `n as the power of a prime;


N = 3, 4, 5 falls into this category but is not assumed. As always, F is a number

field.

Choose j0 ∈ F a different from 0 or 1728, and let E be an elliptic curve with

j-invariant j(E) = j0. Up to a quadratic twist, E has a Weierstrass equation in the

form

y2 = x3 − 3

122

j0j0 − 1728

x− 2

123

j0j0 − 1728

(4.21)

In particular, E has a model which can be defined over F (j0). It will be implicitly

assumed that every elliptic curve is defined over a field F(j(E)

)⊆ F a.

There are N2 points of order N ; in fact, E[N ] ∼= Z/NZ⊕ Z/NZ. Any Galois au-

tomorphism σ ∈ GF (j(E)) permutes the N -division points since [N ] (σP ) = σ([N ]P ),

so in particular the Galois group acts on the `-adic vector space

V`(E) =(lim←−E[`n]

)⊗Z`

Q`∼= Q2

` (4.22)

as associated with the Tate module. The `-adic representation of the elliptic curve

is defined as ρE, ` : GF (j(E)) → GL(V`(E)

)so that the L-series of the elliptic curve is

defined as the L-series associated with the `-adic representation:

Lp (E, T ) = Lp (ρE, `, T ) for p ⊆ F(j(E)

)not dividing `; (4.23)

where we define the conductor of the elliptic curve fE = fρE, `to be the conductor of

the `-adic representation. Note that it depends only on the action of the ramification

groups on V`(E). (The usual definition for the conductor for the elliptic curve differs

from this one by the factors at p dividing `.)

The following proposition shows that this is the usual definition of the L-series

attached to an elliptic curve.

Proposition 4.4.1 (`-adic L-Series of the Elliptic Curve). Let F be a number

field, and E be an elliptic curve with j(E) ∈ F a. Define Lp(ρE,`, T ) as in (4.23) for

primes p - ` in F(j(E)

).

1. If E has good reduction at p, then


Lp(E, T ) = 1− a(p)T + N pT 2 where a(p) = N p + 1− |E(Fp)|. (4.24)

2. Otherwise

Lp(E, T ) =

1− T if E has split multiplicative reduction at p;

1 + T if E has non-split multiplicative reduction at p;

1 if E has additive reduction at p.

(4.25)

In particular, the usual definition for the L-series L(E, s) and the L-series for the

`-adic representation L(ρE,`, s) differ at most by the Euler factors at p dividing `.

Proof. Denote the field fixed by the kernel of ρE, ` as L =∏

n F (E[`n]). The criterion

of Neron-Ogg-Shafarevich (as described in Serre and Tate [ST68]) states that the

following are equivalent for a prime p ⊆ F(j(E)

)not dividing `: 1) p is ramified in L;

2) an elliptic curve E has bad reduction at p; and 3) Tp acts on V`(E) nontrivially. An

elliptic curve has three reduction types, so we have three cases to consider. Denote

E as the reduced curve for the residue field Fp = OF (j(E))/p. The ideas below follow

the exposition in Silverman [Sil94].

Case 1: Good Reduction. The Euler factor is in the form

Lp(E, T ) = 1− tr ρE, `(p)T + det ρE, `(p)T 2 (4.26)

The criterion of Neron-Ogg-Shafarevich states that Tp has trivial action on V`(E)

so that ρE, `(p) = ρE, `(σP). The Weil pairing tells us that det ρE, `(σP) = N p and

det [I − ρE, `(σP)] = |E(Fp)| so that we compute

a(p) = tr ρE, `(σP) = 1+det ρE, `(σP)−det [I − ρE, `(σP)] = N p+1−|E(Fp)| (4.27)


Case 2: Multiplicative Reduction. We have bad reduction at p, so Tp acts on V`(E)

nontrivially. [Sil94] states the general fact

V`(E)Tp ∼=(lim←− Ens[`

n])⊗Z`

Q` (4.28)

where Ens(Fp) is the collection of nonsingular points on the reduced curve E. In

particular, when E has multiplicative reduction, Ens(Fp) ∼= (Fp)× is the multiplicative

group, so V`(E)Tp ∼= Q`. This gives Lp(E, T ) = 1− tr ρE, `(σP)T , and

tr ρE, `(σP) = N p + 1− |E(Fp)| =

+1 for split multiplicative reduction;

−1 for non-split multiplicative reduction.

(4.29)

Case 3: Additive Reduction. In this case Ens(Fp) ∼= (Fp)+ is the additive group,

so V`(E)Tp ∼= {0}. This gives Lp(E, T ) = 1.

In the next chapter we specialize to the case where N is 3, 4, or 5. We find that

given a certain class of complex Galois representations, we can associate a family of

elliptic curves such that the base change of the complex representation is congruent

to a twist of the `-adic representations.

Chapter 5

Representations of Platonic Type

5.1 Introduction

Conjecture 5.1.1 (Artin). Let F be a number field, and ρ : GF → GL(V ) be a

finite dimensional complex representation, not containing the trivial representation,

with finite projective image. Then the associated L-series L(ρ, s) is an entire function.

This conjecture was shown to be true when V ∼= C is 1-dimensional, so in this

chapter we turn our attention to the 2-dimensional space V ∼= C2. We continue to

let F be a number field, but specialize to ρ : GF → GL2(C) not containing the

trivial representation. Klein [Kle93] showed that the only subgroups of PGL2(C) are

isomorphic to either the rotations of regular polygons (Zn or Dn) or the rotations of

regular polyhedra (A4, S4, or A5). Nontrivial complex representations with projective

image isomorphic to the former are induced representations from 1-dimensional ones,

so it follows from classical results due to Hecke that the associated L-series are entire.

In this chapter we study the properties of the latter. Unless noted otherwise, we

assume that N is 3, 4, or 5.

5.2 Associated Elliptic Curves

The rotation groups of the Platonic solids are isomorphic to PSL2 (Z/N Z) for N

being 3, 4, or 5, so we consider Galois representations ρ : GF → GL2(C) such that

59

CHAPTER 5. REPRESENTATIONS OF PLATONIC TYPE 60

im(ρ) ∼= PSL2 (Z/N Z). We give such representations a name.

Definition 5.2.1 (Representations of Platonic Type). Let F be a number field,

V ∼= C2, and ρ : GF → GL(V ) be a representation. We say that ρ is of Platonic

Type if the projective image im(ρ) is isomorphic to the rotations of one of the regular

polyhedra (and√−1 ∈ F if it is isomorphic to S4.)

Given a representation ρ of Platonic Type, denote K as the field fixed by ker(ρ).

Then K/F is a PSL2 (Z/N Z)-extension by definition, so there is an elliptic curve E

defined over F a so that the following diagram holds:

K K(j(E)

)K0 = K(E[N ]x)∣∣∣∣ ∣∣∣∣ ∣∣∣∣

F F(j(E)

)F0 = F

(j(E), ζN

) (5.1)

where K0 = K ·F0 and each of the vertical lines is an PSL2 (Z/N Z)-extension. Using

these ideas, we make a more general definition.

Definition 5.2.2 (Associated Elliptic Curves). Let F be a number field, V be a

finite dimensional complex vector space, ρ : GF → GL(V ) be a representation with

finite projective image, and K be the field fixed by ker(ρ).

Denote Eρ as the set of all elliptic curves E defined over F a such that, for some

positive integer N ,

1. F(j(E)

)is contained in a biquadratic extension of F ;

2. K · F (j(E), ζN) = F (E[N ]x); and

3. K ∩ F (j(E), ζN) = F .

Such an elliptic curve E ∈ Eρ is said to be associated to ρ.

We spend the rest of this chapter understanding the properties of this family.

Proposition 5.2.3. Given ρ and E ∈ Eρ associated to ρ, the following hold:

1. im(ρ) ∼= Gal (F (E[N ]x) /F (j(E), ζN) )


2. im(ρ) ∼= im(ρ|G) where G = GF (j(E)).

3. Eσ is also associated to ρ for any σ ∈ GF .

4. Every representation of Platonic Type has an associated elliptic curve.

Proof. Denote K0 = F (E[N ]x), F0 = F (j(E), ζN), and j0 = j(E). Then by the

First Isomorphism Theorem for groups, im(ρ) = Gal(K/F ) is the same as

Gal(K/F ) = Gal(K /K ∩ F0 ) ∼= Gal(K · F0 /F ) = Gal (F (E[N ]x) /F (j0, ζN) )

(5.2)

The inclusion K ∩ F (j0) ⊆ K ∩ F0 = F shows that F = K ∩ F (j0) as well.

K(j0) = K · F (j0) so im(ρ) = Gal(K/F ) is the same as

Gal(K /K∩F (j0) ) ∼= Gal(K ·F (j0) /F (j0) ) = Gal(K(j0) /F (j0) ) = im(ρ|G) (5.3)

The group GF0 acts trivially on E (since E is defined over F0) so we consider the

quotient Gal(F0/F ). (Note that F (j0)/F is Galois, so F0/F is Galois as well.) Any

σ ∈ Gal(F0/F ) acts trivially on K since K ∩ F0 = F , and j(E)σ = j(Eσ) so that

F(j(Eσ), ζN

)= F σ

0 = F0. K0 is generated by the roots of the N division polynomial,

which has coefficients in F0. Hence σ acts on the coefficients of this polynomial, and

so F (Eσ[N ]x) = Kσ0 . But this is the same as Kσ

0 = Kσ · F σ0 = K · F0 = K0 so the

result follows.

Now let ρ : GF → GL(V ) be a complex Galois representation with im(ρ) ∼=PSL2 (Z/N Z). DenoteK as the field fixed by ker(ρ) so thatK/F is a PSL2 (Z/N Z)-

extension. Theorem 3.4.1 states that there is an elliptic curve E defined over a

biquadratic extension F(j(E)

)such that F (E[N ]x) = K · F0, F = K ∩ F0, and

im(ρ) = Gal(K/F ) ∼= Gal(K0/F0).


5.3 Complex Representations of Z · SL2 (Z/N Z)

When ρ is a representation of Platonic Type, we can associate a family of elliptic

curves Eρ. Conversely, given a family of elliptic curves {E} defined over F a such that

F (E[N ]x) /F(j(E), ζN

)is a PSL2 (Z/N Z)-extension, we may associate a represen-

tation of Platonic Type. Consider the composition of maps

ρE, N : GF (j(E))

ρE, `−−−→ GL2(Z`)mod N−−−−→ GL2 (Z/N Z) for N = `n. (5.4)

ρE, N is also defined as the representation coming from the action of GF (j(E)) on

E[N ] ∼= Z/N Z⊕ Z/N Z. ρE, N is commonly known as the mod N representation.

In order to define a complex Galois representation, we first consider a representa-

tion GL2 (Z/N Z) → GL2(C). Unfortunately, such a map is faithful only if N = `n

is not more than 6, and even then this condition is not sufficient; we must work with

a subgroup of the domain.

Proposition 5.3.1. Set N = 3, 4, or 5. Let ωN be a primitive Dirichlet character

modulo N , Z denote the diagonal matrices of GL2 (Z/N Z) with determinant ±1, and

set ϕN : Z · SL2 (Z/N Z)→ GL2(C) as the map defined by

(−1

1

)7→

(−1

1

),

(a

d

)7→

(ωN(a)

ωN(d)

),

(1 1

1

)7→

(α −ββ α

);

(5.5)

where

α =1

2·

−1 N = 3;

0 N = 4;

−1±√

5

2N = 5.

β = −1

2+i

2·

√

2 N = 3;√

3 N = 4;

1±√

5

2N = 5.

(5.6)

Then ϕN is a faithful representation, and there is a primary ideal λN ⊆ Qa dividing


N such that ϕN ≡ I (mod λN).

When N is 3 or 4, Z · SL2 (Z/N Z) is the full group GL2 (Z/N Z). When N = 5,

the group Z · SL2(Z/5 Z) has index 2. The reason for this choice of subgroups will

be discussed in the proof below.

Proof. Lemma 3.2.2 states that the group Z · SL2 (Z/N Z) is generated by the three

matrices

S =

(1 1

1

), T =

(−1

1

), and

(a

d

); (5.7)

where a d ≡ ±1 (N), so it suffices to define ϕN on these generators. We begin by

defining the image of the latter two matrices as described above.

To find the image of the third generator S, we express ϕN(S) as a unitary ma-

trix with determinant 1. By direct computation, one verifies that (T S)3 = −I.Then the image ϕN (T S)3 = −I as well, so the trace of ϕN (T S) must be equal

to 2 Re(ζ6); that is −2 Re(β) = 1. The conjugate T S T−1 = (S−1)t is the trans-

pose of the inverse, so the image should have the same property. ϕN(S) is a unitary

matrix, and so the inverse is simply the transpose of the complex conjugate. That

is, ϕN(T )ϕN(S)ϕN(T )−1 = ϕN(S) so we find that α = α. Using the fact that S

has order N , we conclude that α = Re(ζN). The final condition detϕN(S) = 1

determines α and β as above. (One finds the imaginary part of β by the relation

4(Im(β)

)2= 1− 2 Re(ζ2

N). Both sides are positive only if N < 6 so the condition N

= 3, 4, or 5 is indeed necessary.)

We must verify that these definitions do yield a bona fide complex representation.

We follow the exposition in Bump [Bum97] on representations of SL2(Fq) by consid-

ering various conjugation relations among the generators. Although Bump considers

finite fields, it is not difficult to generalize to Z/N Z for N being the power of a prime.

The first of the relations is immediate from the choice of definition:


T

(a

d

)T−1 =

(d

a

)=⇒ T

(ωN(a)

ωN(d)

)T−1 =

(ωN(d)

ωN(a)

)(5.8)

while the second relation is

(a

d

)S

(a

d

)−1

= Sn =⇒

(α −β ωN(n)

β ωN(n) α

)=

(α −ββ α

)n

(5.9)

where n ≡ a d−1 (mod N). It is easy to see that this condition is satisfied if n ≡ ±1;

recall that the inverse of ϕN(S) is the transpose of the complex conjugate, and that

ωN(−1) = −1. But this is always the case since matrices in Z satisfy a d−1 ≡ ±a d ≡±1 (N). (Note that this condition is not satisfied if N = 5 and n ≡ ±2 (5). This

explains why we must restrict to this subgroup.)

The third relation states

T Sn T−1 =

(m

n

)S−n T−1 S−m or (T Sn) (T Sm) (T Sn) =

(−m

−n

)(5.10)

where mn ≡ 1. We must check that this relation holds for ϕN(T Sn) for each n ∈(Z/N Z)×. However, once we verify that this true for n we may take inverses of both

sides to find that it is true for −n as well; this limits half of the work. By construction,

ϕN(T S)3 = −I so the equation holds for n ≡ 1. The only remaining case is N = 5

and n ≡ 2 (5), which one may show directly:

ϕ5

(T S2

)ϕ5

(T S3

)ϕ5

(T S2

)=

(−ω5(3)

−ω5(2)

)(5.11)

Hence, all three relations hold for the matrices in the image, so ϕN is indeed a bona

fide complex representation of Z · SL2 (Z/N Z).

We now prove the statement about the primary ideal. Each of the coefficients


α, β, and ωN(a) lie in the field Q(Re(ζN),

√1−N

), where α, β lie in its ring of

half-integers. Consider the ideal λN generated by 2 (1−α), 2 β, and a−ωN(a) for all

a ∈ Z/N Z. Explicitly,

λN is the primary ideal generated by

−1 +

√−2 for N = 3;

2 for N = 4;√

5 and 2− ω5(2) for N = 5.

(5.12)

By construction, λN lies above N and we have the congruence ϕN ≡ I (mod λN).

5.4 Complex Representations from Elliptic Curves

Let E be an elliptic curve over F a such that ρE, N is a representation intoGL2 (Z/N Z).

We would like to consider the composition ϕN ◦ ρE, N as a complex representation,

but this composition makes sense only if im(ρE, N) is contained in Z · SL2 (Z/N Z).

This is always the case if N is 3 or 4; we need√

5 ∈ F(j(E)

)for N = 5. The

results of chapter 2 show that every elliptic curve in Eρ associated to an icosahedral

representation satisfies this criterion.

Given a representation ρ of Platonic Type, we now define a complex L-series for

each E ∈ Eρ as that L-series attached to the mod N representation:

Lp

(ρE, N , T

)= det

[I −

(ϕN ◦ ρE, N

)(p) T

]for p ⊆ OF (j(E)) not dividing N .

(5.13)

Proposition 5.4.1 (Mod N L-Series of the Elliptic Curve). Let F be a number

field, ρ be of Platonic Type, and E ∈ Eρ associated to ρ. Let N be 3, 4, or 5 if ρ

is tetrahedral, octahedral, or icosahedral, respectively; and define Lp(ρE, N , T ) as in

(5.13) for primes p - N in F(j(E)

).

1. If E has good reduction at p, then there is a primary ideal λN | N in the algebraic

closure F a such that


Lp

(ρE, N , T

)= 1− a(p)T + ωN (N p) T 2 where a(p) ≡ a(p) (λN). (5.14)

2. Otherwise

Lp

(ρE, N , T

)=

1− T if E has split multiplicative reduction at p;

1 + T if E has non-split multiplicative reduction at p;

1 if E has additive reduction at p.

(5.15)

3. L(E, s) ≡ L(ρE, N , s

)(mod λN).

4. A prime divides the conductor of the mod N representation if and only if it

divides the conductor of E.

The coefficients a(p) may be calculated by considering the order of the Frobenius

element σP. Here are the different cycle types in PSL2 (Z/N Z) for N = 3 or 5:

Cycle Type 1 2,2 3 N

Matrix I T T S S or S2

Trace 2 0 -1 2 Re ζN or 2 Re ζ2N

a(p)2 / ωN(N p) 4 0 1 (2 Re ζN)2 or (2 Re ζ2N)

2

(5.16)

Assuming that the elliptic curve E comes from some polynomial q(x) defined over F ,

the cycle type may be determined by factoring this polynomial modulo primes p in

F(j(E)

). The coefficients a(p) may be determined up to sign via the table above,

and the sign may be determined via the congruence a(p) ≡ N p + 1− |E(Fp)| (λN).

Proof. The criterion of Neron-Ogg-Shafarevich states that, for a prime p ⊆ OF (j(E))

not dividing N , an elliptic curve E has bad reduction at p if and only if Tp acts

on E[N ] nontrivially. As in the proof of Proposition 4.4.1, we have three cases to

consider.


Case 1: Good Reduction. The Euler factor Lp

(ρE, N , T

)is in the form

1− tr[ϕN ◦ ρE, N(σP) ερE,`

(p)]T + det

[ϕN ◦ ρE, N(σP) ερE,`

(p)]T 2 (5.17)

First, we calculate det(ϕN ◦ ρE, N

)(p). The map detϕN only depends on the

scalar matrices in Z · SL2 (Z/N Z):

det(ϕN ◦ ρE, N

)(p) = det

(ωN(a)

ωN(d)

)= ωN

(det

(a

d

))= ωN (det ρE, `(p)) = ωN (N p)

(5.18)

Second, we find the congruence relation with the trace. Proposition 5.3.1 states

that ϕN is the identity modulo some primary ideal λN , so we have

a(p) = tr(ϕN ◦ ρE, N

)(p) ≡ tr ρE, `(p) ≡ a(p) (mod λN) (5.19)

Case 2: Multiplicative Reduction. We have bad reduction at p, so V`(E)Tp ∼= Q`

and Lp

(ρE, N , T

)= 1− tr

(ϕN ◦ ρE, N(p)

)T . The inertia group also acts nontrivially

on E[N ]. Only one of the generators is ramified, so the nonsingular points form

a cyclic group of order N ; the automorphisms of this group must be contained in

(Z/N Z)× ∼= Z. We know that tr ρE, `(p) = ±1 depending on whether the reduction

is split or non-split. The only diagonal matrix in Z with ±1 as an eigenvalue is ±I.Hence, tr

(ϕN ◦ ρE, N

)(p) = tr

(ϕN(±I) ερE,`

(p))

= ±1.

Case 3: Additive Reduction. The idempotent ερE, `(p) is trivial, so the Euler factor

is trivial as well.

The statement about the conductors follows from the fact that the action by the

inertia groups on the `-adic representation (i.e. V`(E)) is the same as the action on

the mod N representation (i.e. E[N ]). Any prime that divides the conductor of E

has nontrivial action of V`(E) and so has nontrivial action on E[N ].


5.5 Congruences Among L-Series

The statements above may be combined to make a strong conclusion about the family

Eρ when ρ is of Platonic Type.

Theorem 5.5.1. Let F be a number field, and ρ : GF → GL(V ) be of Platonic Type.

Then for each associated elliptic curve E ∈ Eρ there exists a Hecke character χ and

a primary ideal λ in the algebraic closure F a such that

1. λ lies above N , where N = `n is 3, 4, or 5 if ρ is tetrahedral, octahedral, or

icosahedral, respectively.

2. L(ρ|G, s

)≡ L

(E, χ, s

)(mod λ) as L-series over F

(j(E)

)where G = GF (j(E)).

3. χ2 = εχ ◦ NF where εχ = ωN ⊗ det ρ.

4. a(pσ) ≡ Θσ(p) a(p) (mod λ) where Θσ(p) = χ(pσ−1) is a quadratic character

for each σ ∈ GF .

If a prime divides the conductor of ρ then it divides the conductor of E as well.

The Hecke character χ in this theorem shows that E is “almost” isogenous to

a twist of its conjugate. Indeed, a(pσ) ≡ Θσ(p) a(p) for some quadratic character,

although we may not have equality.

Proof. Let ρ be of Platonic Type and E ∈ Eρ be an associated elliptic curve. Denote

ρ1 = ρ|G as the restriction to the subgroup G = GF (j(E)), and ρ2 = ϕN ◦ ρE, N as the

complex representation coming from the mod N representation of the elliptic curve.

If we set L as the field fixed by ker(ρ1), K as the field fixed by ker(ρ1), and V ∼= C2,

then the following diagram shows the strategy for the following proof:

1 −−−→ Gal(L/K) −−−→ Gal(L/F ) −−−→ Gal(K/F ) −−−→ 1yχ

yρ1

yρ1

1 −−−→ Z −−−→ GL(V ) −−−→ PGL(V ) −−−→ 1

(5.20)


First, we show equality among the maps on the rightmost column: We claim that

we may choose ϕN so that the projective representations ρ1 and ρ2 are equal up to

conjugation. Indeed, there are at most two projective complex representations of

PSL2 (Z/N Z) which differ by the trace of the elements of order N , and the base

change ρ1 = ρ|G is a representation of Platonic Type because proposition 5.2.3 states

that im(ρ) ∼= im(ρ|G). If ρ1 and ρ2 are not the same, then choose α = Re(ζ2N) instead

of α = Re(ζN) in the definition for ϕN(S) in proposition 5.3.1.

Second, we show that there exists a map χ in the leftmost column: We claim that

any two projective representations which are equal must be scalar multiples of each

other by some Hecke character. We’ve just seen that ρ1 = ρ2, so define the elements

χ(σ) by the relation ρ1(σ) = χ(σ) ρ2(σ) for each σ ∈ G. It is immediate that χ is

actually a multiplicative character since, given any σ and τ in G,

χ(σ τ) = ρ1(σ τ) ρ2(σ τ)−1 = ρ1(σ) ρ1(τ) ρ2(τ)

−1 ρ2(σ)−1 = ρ1(σ) ρ2(σ)−1 χ(τ)

= χ(σ)χ(τ)

(5.21)

Recall that we may interpret the 1-dimensional representation χ : G→ C× as Hecke

character by considering the Frobenius elements.

This shows that ρ|G = χ ⊗(ϕN ◦ ρE, N

)— up to conjugation — so in particular

ρ|G ≡ χ ⊗ ρE, ` (mod λN) as in proposition 5.3.1. The statement about the L-series

follows. Upon considering the determinant of this relation, we have det ρ ◦ NF =

χ2 ⊗ (ω5 ◦ N).

For each σ ∈ GF , consider the ratio

a(pσ)2

ωN(N p)=

(tr ρ(p))2

det ρ(p)=

a(p)2

ωN(N p)=⇒ a(pσ)2 = a(p)2 (5.22)

Then the ratio Θσ(p) = χ(pσ−1) = a(p)/a(pσ) = ±1 is indeed a quadratic character.

Hence a(pσ) = Θσ(p) a(p) gives the desired result.

A prime divides the conductor of the base change ρ|G if and only if it divides the

conductor of the mod N representation if and only if it divides the conductor of E.


If a prime divides the conductor of ρ, then it either divides the conductor of the field

F(j(E)

)or the conductor of the base change ρ|G.

In the next chapter, we will see this theorem at work: We will study a specific

example of an icosahedral representation ρB : GQ → GL2(C), and list the properties

of a specific associated elliptic curve EB ∈ EρBwith remarkable properties. The

character χ will be of central importance.

Chapter 6

Deformations of Buhler’s Modular

Form

6.1 Introduction

In the previous chapter we proved that representations ρ of Platonic Type (i.e. com-

plex Galois representations with projective image isomorphic to the rotations of one

of the regular polyhedra; see definition 5.2.1) have associated elliptic curves E ∈ Eρ

(see definition 5.2.2) such that the L-series of the base change of ρ is a Galois defor-

mation of the twist of the `-adic representation. That is, there is some primary ideal

λ ⊆ Qa and some Hecke character χ such that

L(ρ|G, s

)≡ L

(E, χ, s

)(mod λ) where G = GF (j(E)). (6.1)

In [Buh78], the author exhibits an irreducible icosahedral representation ρB at-

tached to a weight 1 modular form, therefore exhibiting one of the first examples to

verify Artin’s conjecture. In this chapter, we show that this representation is actually

a semisimplification of the twist of a complex representation coming from a modular

elliptic curve EB defined over Q(√

5).

71

CHAPTER 6. DEFORMATIONS OF BUHLER’S MODULAR FORM 72

6.2 Principal Quintics and Elliptic Curves

In [Buh78], we are introduced to the quintic

qB(z) = z5 + 10 z3 − 10 z2 + 35 z − 18 (6.2)

with discriminant Disc (qB) = 26 58 112. Let Q(qB) denote its splitting field. Since qB

has degree 5, we know that Gal( Q(qB)/Q ) is a subgroup of S5, the symmetric group

on 5 letters. The discriminant is a square, so the Galois group is actually contained

in the alternating group A5. We have the factorizations

qB(z) ≡

z (z + 2) (z3 + z2 + 2 z + 1) (mod 3)

z5 + 3 z3 + 4 z2 + 3 (mod 7)

(z − 3) (z2 − 2 z − 2) (z2 + 5 z − 3) (mod 11)

(6.3)

so that the Galois group contains a 2,2-cycle, a 3-cycle and a 5-cycle. We conclude

that Gal( Q(qB)/Q ) is a subgroup of A5 of index dividing 2. Hence, qB(z) generates

an A5 extension.

In chapter 2, we found that we can always transform a quintic into a principal

quintic by way of a Tschirnhaus transformation. (See lemma 2.3.3.) The Tschirnhaus

discriminant for the quintic above is Tsch(qB) = 12500 = 24 55, so we expect to find

a principal quintic with coefficients in the quadratic extension Q(√

5). Indeed, using

those techniques we find the principal quintic

θ(x) = z5 − −1 +√

5

2150 z2 +

−5 + 13√

5

225 z + (−132 + 47

√5) 50 (6.4)

which has the associated j-invariant

j0 =3763709089861481711841390880

1963436211517444265962051− 3711313749236962322121473696

1963436211517444265962051

√5 (6.5)

The denominator is 722115, so we will have bad reduction at this prime. We


present an alternate method for constructing the elliptic curve since we would like to

have bad reduction only at primes above 2 and 5.

Proposition 6.2.1. Let qB(z) = z5 + 10 z3 − 10 z2 + 35 z − 18 have splitting field

Q(qB). Then for any elliptic curve E defined over Q(√

5) with j-invariant jB =

86048− 38496√

5, we have Q(EB[5]x) = Q(qB, ζ5), and

Gal (Q(qB)/Q) ∼= Gal(

Q(θB)/Q(√

5))∼= Gal ( Q(EB[5]x)/Q(ζ5) ) ∼= A5 (6.6)

where Q(θB) is the splitting field of the principal quintic

θB(z) = z5 − 125(185 + 39

√5)z2 − 6875

(56 + 19

√5)z − 625

(10691 + 2225

√5).

(6.7)

Proof. We begin by showing that Q(qB,√

5) is the splitting field of the principal

quintic θB. Introduce a fractional linear transformation so that

qB(w) =a5 z

5 + a4 z4 + a3 z

3 + a2 z2 + a1 z + a0

(c z + d)5where w =

a z + b

c z + d. (6.8)

We wish to set a4 = a3 = 0. The coefficient a4 is given by

a4 = 5 b(a4 + 6 a2 c2 − 4 a c3 + 7 c4

)− 5 d

(−4 a3 c+ 6 a2 c2 − 28 a c3 + 18 c4

)(6.9)

We compute that a4 = 0 when

b = −4 a3 c+ 6 a2 c2 − 28 a c3 + 18 c4 and d = a4 + 6 a2 c2 − 4 a c3 + 7 c4 (6.10)

This choice of b and d forces a5 = a d−b c and a3 = a5 (a2−a c−c2) (10 a4−20 a3c+. . . ).

To set a3 = 0 we will let a2 − a c − c2 = 0. These equations are satisfied when


a = (1 +√

5)/2, b = 5− 15√

5, c = 1 and d = (35 + 5√

5)/2. Then (c z + d)5 qB(w) =

(a d − b c) θB(z) where θB is the monic polynomial stated in the proposition. It has

discriminant Disc(θB) = (ad− bc)12 Disc(qB), which is also a square in Q(√

5).

The map z 7→ w shows that the splitting field of θB is Q(θB) = Q(qB,√

5) since the

roots of qB can be mapped is this way to the roots of θB. Clearly Gal( Q(θB)/Q(√

5) )

has index dividing 2 in A5. We must have equality since A5 has no nontrivial sub-

groups of this type.

The statement about the elliptic curve follows from corollary 2.5.2 since one com-

putes that θB(z) has associated j-invariant jB = 86048− 38496√

5.

6.3 The Elliptic Curve EB

We may exhibit a particular Weierstrass equation with j-invariant jB as in the propo-

sition above such that discriminant equal to a power of 2 times a power of 5. Such

an elliptic curve can be derived by first choosing a Weierstrass equation for the j-

invariant (as in (2.34)) then finding a global minimal model for this equation. Such

a model exists since Q(√

5) has class number 1.

One which works is given by

EB : y2 = x3+(5−√

5)x2+√

5x with j-invariant jB = 86048−38496√

5. (6.11)

The discriminant is ∆ = 2400 − 1120√

5 = −ε4 26√

53, where ε = −1+

√5

2= 2 Re(ζ5)

is the fundamental unit for the integral closure of Q(√

5).

Proposition 6.3.1. The elliptic curve EB defined in (6.11) has Type II reduction at

p2 = (2) with exponent 6 and Type III at p5 = (2 − ε) with exponent 2, and Type

I0 reduction at all other primes. Moreover, EB is in global minimal form and has

conductor fB = (320).

In particular, the elliptic curves E with j-invariant jB are not semistable since

they have additive reduction at the primes above 2 and 5.


Proof. This curve has discriminant (∆) = p62 p3

5, so there is bad reduction at the

primes dividing 2 and 5. We will use Tate’s Algorithm to find more information

about such primes. (For a discussion of the algorithm, consult [Sil94].)

In order to use Tate’s Algorithm, we first move the singular point on EB to the

origin. Such a point comes from any repeated roots of the cubic x3 + a x2 + b x when

reduced modulo a prime, so express this polynomial as

x3 + (5−√

5)x2 +√

5x

=(T −√

5 ε3) (

T 2 + 2√

5 ε2 T + 2√

5 ε3) where T = x+

√5 ε3.

(6.12)

Set this polynomial as P (T ) = T 3 + a2 T2 + a4 T + a6, and consider the ideals

(a2) = p25, (a4) = p2 p5, (a6) = p2 p2

5; =⇒ (b2) = p22 p2

5, (b8) = p22 p2

5 p11; (6.13)

where p11 = (3− ε) is one of the primes above 11.

First consider the prime p2. It divides (∆), and (b2), but p22 does not divide (a6).

By Tate’s Algorithm we have Type II reduction. The exponent of the conductor is

f = ord2(∆) = 6.

Now consider the prime p5. It divides (∆) and (b2), while p25 divides (a6), but p3

5

does not divide (b8). By Tate’s algorithm we have Type III reduction. The exponent

of the conductor is f = ord5(∆)− 1 = 2.

The algorithm terminates so EB must be in global minimal form.

6.4 The Hecke Character χB

We begin this section with a special property of this j-invariant.

Proposition 6.4.1. Let E be an elliptic curve over Q(√

5) with j(E) = jB. Then

E is isogenous to a twist of its conjugate; that is, E is a quadratic Q-curve. In


particular, EB is 2-isogenous to a twist of its conjugate by the quadratic character(−2·

)◦ N.

A Q-curve is an elliptic curve E defined over Qa which is isogenous over Qa to all

of its conjugates Eσ. A quadratic Q-curve is one which is defined over a quadratic

extension of Q. (This definition may be found in Ribet [Rib92].) This proposition

is significant in that we are able to relate some of the ideas in [Buh78] with some

of the more recent trends regarding elliptic curves, most notably the classification in

[GL98].

Proof. There exists d ∈ Q(√

5)× such that E has the Weierstrass model

E : y2 = x3 + a x2 + b x where a = (5−√

5) d, b =√

5 d2. (6.14)

Set µ = ε/dσ ∈ Q(√

5), and consider the elliptic curve

E ′ : Y 2 = X3 +AX2 +BX where A = −2 a µ−2, B = (a2−4 b)µ−4. (6.15)

E and E ′ are 2-isogenous via the map φ : E → E ′ given by

φ : (x, y) 7→(µ2 y

2

x2, µ3 y (b− x2)

x2

); µ =

ε

dσ∈ Q(

√5). (6.16)

One calculates that j(E ′) = j(E)σ, given that σ is the nontrivial automorphism

of Gal(Q(√

5)/Q). Moreover, A = aσ D and B = bσ D2 where D = −2 · N d, given

that N : Q(√

5) → Q is the norm map. Hence E ′ is a twist of Eσ by the quadratic

character(

D·

)◦ N. (The character factors through the norm map because D is an

element of Q.)

This proves that E is isogenous to a quadratic twist of Eσ. The statement about

EB follows from setting d = 1.

We consider the elliptic curve E defined in (6.14) in more detail. First we prove

a general result about elliptic curves isogenous to quadratic twists of each other.


Lemma 6.4.1. Let E1 and E2 be two elliptic curves defined over a number field F

such that E1 is isogenous to a twist of E2 by the quadratic character(

D·

). Then for

every prime p not dividing D or either conductor fE1 or fE2,

a1(p) =

(D

p

)a2(p) where ai(p) = N p + 1− |Ei(Fp)|. (6.17)

Proof. Denote p as a prime in F and P as a prime in F (√D) lying above p. Let ρEi,` :

GF → GL2(Q`) be the `-adic representation; and αi, βi be the distinct eigenvalues of

the image ρEi,`(σP). Then

ai(p) = αi+βi, N p = αi βi =⇒ |Ei(Fp)| = 1−ai(p)+N p = (1− αi) (1− βi) (6.18)

E1 and E2 are isogenous over F (√D) so that

(1− αf

1

) (1− βf

1

)= |E1(FP)| = |E2(FP)| =

(1− αf

2

) (1− βf

2

)(6.19)

which implies a1(P) = a2(P). Hence, when p splits i.e.(

Dp

)= 1 we conclude that

a1(p) = a2(p). We now consider the case when p is inert i.e.(

Dp

)= −1.

Let A = ResF (√

D)/FE2 be the restriction of scalars of the curve E2 as viewed over

F (√D). Then A is isogenous to E1⊗E2. Counting points as curves over FP, we have

(1− α2

2

) (1− β2

2

)= |A(FP)| = |E2(FP)| = |E1(Fp)| · |E2(Fp)|

= (1− α1) (1− β1) · (1− α2) (1− β2)(6.20)

Upon canceling common factors, we find (1− α1) (1− β1) = (1 + α2) (1 + β2), which

implies a1(p) = α1 + β1 = −(α2 + β2) = −a2(p). This gives the desired result.

When E is an elliptic curve over Q(√

5) with j(E) = jB as defined in (6.11), propo-

sition 6.4.1 shows that E is isogenous to a twist of its conjugate Eσ by a quadratic


character(

D·

)for some D ∈ Q. If there were a Hecke character χ defined on Z[ε]

such that

χ(nσ) =

(D

N n

)χ(n) where Gal

(Q(√

5)/Q)

= 〈σ〉; (6.21)

then χ(p) a(p) = χ(pσ) a(pσ), and the twisted Euler factor would be independent of

the choice of prime p lying above p:

Lp(E, χ, T ) = 1− χ(p) a(p)T + χ(p)2 N pT 2

= 1− χ(pσ) a(pσ)T + χ(pσ)2 N pT 2 = Lpσ(E, χ, T )(6.22)

We now turn our attention to constructing such a character.

Proposition 6.4.2. Define the character χB : Z[ε]→ C by the relations

χB(n) = (−1)α+γ+δ iβ when n ≡ (−1)α 3β 11γ (1+20 ε)δ εθ (mod 40 Z[ε]) (6.23)

if N n is relatively prime to 40, and χB(n) = 0 otherwise. Then χB has the following

properties:

1. χB(−1) = −1 and χB(ε) = 1. That is, χB is Hecke character defined on the

ideals of Q(√

5).

2. χB(n) =

(2

n

)ω5(n) for rational integers n, where ω5 : 2 7→ i is the Dirichlet

character modulo 5. That is, when restricted to Z, χB is a Dirichlet character

modulo 40.

3. χB(nσ) =

(−2

N n

)χB(n) and χB(n)2 = εχ(N n), where εχ =

(−1

·

)⊗ ω5.

Proof. We begin by considering the character(−2

·

). It is defined on the ring Z/8 Z,

and so it has conductor 8. Now consider the quotient ring Z[ε]/n where 8 divides

n. We will define the character χB by considering the generators of the invertible


elements in this quotient ring, so we first calculate the order of this group. We state

a more general yet classical result.

Lemma 6.4.3 (Generalized Euler Totient). Let F be a number field and n be a

nonzero ideal in the ring of integers OF . Then

∣∣(OK/n)×∣∣ = N n

∏p|n

(1− 1

N p

)(6.24)

To see why, first assume that n = pe is the power of a prime. An integer n ∈ OK

is invertible modulo pe if and only if n /∈ p. There are N pe elements in OK/pe, of

which Npe−1 are elements of the subring p/pe. Hence, there are Npe−Npe−1 invertible

elements modulo pe. This gives

∣∣(OK/n)×∣∣ =

∏p|n

∣∣(OK/pe)×∣∣ =

∏p|n

(N pe − N pe−1

)= N n

∏p|n

(1− 1

N p

)(6.25)

which is the desired formula.

In the case that concerns us, set n = (40). (This choice is justified by the fact

that the conductor of the elliptic curves of interest is divisible by 5.) By the lemma,

the quotient G =(

Z[ε]/(40))×

has 960 elements. In fact, by counting invertible

elements, we find that this group may be decomposed into the two groups

H = 〈±1, 3, 11 | 34 ≡ 112 ≡ 1 〉 and K = 〈 ε, 1 + 20 ε | ε60 ≡ (1 + 20 ε)2 ≡ 1 〉.(6.26)

It is easy to check that H has 16 elements, K has 120 elements, H∩K = 〈ε30 ≡ −11〉has order 2, and that G = HK is the product. We may choose the values of χB on

the generators as we wish, as long as we pay attention to the value of the character

on this intersection.

When restricting to Z, the subgroup H =(

Z/(40))×

is generated by -1, 3, and

11. The symbol (2· ) is a Dirichlet character modulo 8, and the character ω5 : Z→ C

defined by 2 7→ i is the Dirichlet character modulo 5. We define χB on this subgroup


as

χB(n) =

(2

n

)ω5(n) for rational integers n; (6.27)

so that χB(3) = i and χB(−1) = χB(11) = −1. Note that χB(−11) = 1 so it is well

defined in this case.

On the other hand, the subgroup K is generated by ε and 1 + 20 ε. We define

χB on this subgroup to be trivial; that is χB(ε) = 1 and χB(1 + 20 ε) = −1. (We

may choose the image of these generators to be ±1; we make this choice to match

coefficients with the complex icosahedral representation in [Buh78].)

By direct verification on the generators -1, 3, 11, ε, and 1 + 20 ε, one finds

χB(n)2 = εχ(N n) for algebraic integers n, where εχ =

(−1

·

)⊗ ω5. (6.28)

From these two relations, it is clear that χB has order 4 i.e. (χB)4 = 1. This gives

the result

χB(nσ) = χB(nσ)χB(n)4 = χB(N n)χB(n)2 χB(n)

=

[(2

N n

)ω5(N n)

] [(−1

N n

)ω5(N n)

]χB(n) =

(−2

N n

)χB(n)

(6.29)

(Recall that ω5 is a quadratic character when restricted to Q(√

5).)

We now show that this character may be identified as a character on the ideals of

Q(√

5). This field has narrow class number 1, so any ideal n has some totally positive

generator α which is unique up to multiplication by a totally positive unit in the form

ε±2 k. By construction, χB is trivial on the totally positive units i.e. χB(ε)2 = 1, so

χB(n) = χB(α) is well defined.


6.5 Base Change and Modularity

There are some important immediate consequences of the construction of χB from

the previous proposition.

Proposition 6.5.1. Let E be an elliptic curve defined over Q(√

5) with j(E) = jB.

Then there is a Hecke character χ defined over Q(√

5) such that

1. The twisted Euler factor Lp(E, χ, T ) is independent of the choice of prime ideal

p lying above p.

2. The twisted 5-adic representation χ⊗ ρE, 5 is isomorphic to the base change of

a 5-adic representation ρχ, 5 defined over Q with nebentype εχ.

3. The twisted mod 5 representation χ⊗ ρE, 5 is isomorphic to the base change of

a complex representation ρB defined over Q with nebentype εB =(−1

·

).

In other words, the existence of the j-invariant jB predicts the existence of a

complex icosahedral representation over Q. This is the representation studied in

[Buh78].

This theorem is the converse of theorem 5.5.1 in the following sense: Given a

representation ρ of Platonic Type, we can predict the existence of associated elliptic

curves E and Hecke characters χ. In the specific case of EB along with χB, we can

predict the existence of a representation ρB of Platonic Type.

Proof. Let E be the elliptic curve as defined in (6.14), and set D = −2 · N d. Define

the character χ =(N d

·

)⊗ χB. Given any n ∈ Z[ε], we have

(N d

nσ

)=

(N d

nσ

) (N d

n

)2

=

(N d

N n

) (N d

n

)=⇒ χ(nσ) =

(D

N n

)χ(n) (6.30)

We also find that χ(p) a(p) = χ(pσ) a(pσ) and χ2 = εχ ◦ N. Hence, the twisted Euler

factor Lp(E, χ, T ) is independent of the choice of prime ideal p lying above p.

The twisted L-series takes on the form


L(χ⊗ ρE, 5, s) = L(E, χ, s) =∏

p

1

(1− χ(p) a(p) N p−s + χ(p)2 N p1−2s)f

=∏

p

1

(1− aχ(N p) N p−s + εχ(Np) N p1−2s)f= L

(ρχ, 5|G, s

) (6.31)

as the base change of another L-series for the Galois group G = GQ(√

5). The existence

of the 5-adic representation ρχ, 5 follows from results proved by Langlands [Lan80a]

on representations for GL(2). Explicitly, we have set the Euler factors as

Lp(ρχ, 5, T ) = 1− aχ(p)T + εχ(p) p T 2 where εχ =

(−1

·

)⊗ ω5. (6.32)

Now consider the mod 5 representation ρE, 5 associated with E. The twisted

complex representation ρ = χ ⊗(ϕ5 ◦ ρE, 5

), with ϕ5 as defined in proposition 5.3.1,

is the same for all primes p lying over p; i.e. ρ(p) = ρ(pσ). Hence, ρ ∼= ρB|G is the

base change of a complex representation defined over Q. Again, the existence of ρB

follows from results proved by Langlands. Its nebentype εB is given by

det ρ = χ2 ⊗ det(ϕ5 ◦ ρE, 5

)= εB ◦ N where εB = εχ ⊗ ω5 =

(−1

·

). (6.33)

This completes the proof.

The existence of the character χ gives us a simple way to prove the modularity of

the family of curves E.

Theorem 6.5.1. Let E be an elliptic curve defined over Q(√

5) with j(E) = jB.

Then E is modular.

The modularity of the elliptic curve E is deduced from the fact that it is the

twist of the base change of a modular form from Q. This concept may be useful for

Q-curves in future research since the current techniques used to prove modularity


exploit the properties of Q, not a number field such as Q(√

5). This idea will be

explored more in the next chapter; see 7.0.2.

Proof. It suffices to show that E = EB is modular. We must show that there is

an eigenform f with eigenvalues af (p) equal to the traces of Frobenius χB(p) a(p)

associated to the elliptic curve EB for all p - 2, 5. A result of Faltings [Fal83] states

that one only need check finitely many traces to guarantee equality. We closely follow

the exposition in Livne [Liv87] in order to find exactly how many we must check. The

following is a simplified version of his proof.

For the remainder of the proof, fix the following notation: f ∈ S2(Γ0(N fB), εχ)

is a modular form over Q — which may not necessarily be a newform — where

fB = p62 p2

5 as in proposition 6.3.1 and εχ =(−1

·

)⊗ ω5 as in proposition 6.5.1. (We

fix the Dirichlet character ω5 such that 2 7→ i.) We choose f such that af (p) ∈ Q(i).

The map ρf : GQ → GL2

(Q2(i)

)is the continuous 2-adic representation attached to

f via Eichler-Shimura theory; see [Shi94]. Note that ρf is unramified outside of 2 and

5. O = Z2[i] as the ring of integers in Q2(i), and ℘ = (1 + i) is its maximal ideal.

Lemma 6.5.2. Let f ∈ S2(Γ0(N fB), εχ) be an eigenform with eigenvalues af (p) ∈ O.

If af (3) ≡ 0 (℘) then tr ρf ≡ 0 (℘) identically.

Assume without loss of generality that im(ρf ) ⊆ GL2(O), and let ρf : GQ →GL2(O/℘) be the reduction of ρf mod ℘. Let L be the field fixed by ker(ρf ) so

that Gal(L/Q) is a quotient of GL2(O/℘) ∼= S3. Say that tr ρf (σ) = 1 for some

σ ∈ Gal(L/Q); then σ must necessarily have order 3. By results in [Buh78, page

109], L contains a root of q(x) = x3 − x2 + 2x + 2, which, modulo 3, is irreducible.

Hence the Frobenius element σ3 must have order 3 so that af (3) = tr ρf (σ3) ≡ 1, a

contradiction.

Define the representation ρ : GQ(√

5) → GL2

(Q2(i)

)2as ρ = (ρ1, ρ2), where

ρ1 = ρf |GQ(√

5)and ρ2 = χB ⊗ ρE, `. The matrices ρν(σ) have entries in Q2(i), so

replace ρν by a conjugate matrix if necessary so that each ρ(σ) has entries in O.

Lemma 6.5.3. Let f be as in 6.5.2. det ρν ≡ 1 and tr ρν ≡ 0 (℘) for ν = 1, 2. In

particular, ρ2 ≡ I (℘).


Each ρν(σ) is invertible, so its image must be det ρν(σ) = 1. Let p ⊆ Q(√

5) be

a prime lying above a rational prime p. It is clear from 6.5.2 that tr ρ1 ≡ 0. EB

has a rational point of order 2, so a(p) = N p + 1 − |E(Fp)| ≡ 0 for for all primes

p of good reduction. When p | 2, 5 one checks that a(p) = 0. The final statement

follows from the characteristic polynomials; each ρν satisfies ρ2ν − tν ρν + dν I = 0

where tν = tr ρν ≡ 0 and dν = det ρν ≡ 1.

The image G = im(ρ) a pro 2-group, so denote G∗ as its Frattini subgroup (i.e.

the intersection of all maximal ideals of G). Consider the field fixed by the kernel

of the map GQ(√

5) → G/G∗ induced by ρ. It has Galois group G/G∗, an elemen-

tary abelian 2-group, so that it is contained in the compositum Q(√

5){2,5} of all

quadratic extensions of Q(√

5) which are unramified outside of 2 and 5. We have the

isomorphism

Gal(Q(√

5){2,5}/Q(√

5))∼= (Z2)

4 ; σp 7→{(

2

p

),

(2− ε

p

),

(−1

p

),

(ε

p

)}.

(6.34)

(Recall that p2 = (2) and p5 = (2−ε) are the primes lying above 2 and 5, respectively.)

One checks that the subset Σ ⊆ GQ(√

5) given by

Σ = {1} ∪{σp

∣∣ p lies above 3, 11, 19, 29, 31, 41, 61, 79, 89, 101, or 109}

(6.35)

maps surjectively onto Gal(Q(√

5){2,5}/Q(√

5))

and hence surjectively onto G/G∗.

(The explicit image of Σ can be found in the appendix.)

Lemma 6.5.4. Let M = O[G] be the linear O-span of im(ρ). Then M is spanned by

im(ρ|Σ).

Let M = M/℘M , and G = G/℘M . Clearly M is spanned by G, a finite group of

exponent 2 (since ρ2 ≡ I by 6.5.3). Consider the surjective maps Σ→ G/G∗ → G/G∗.

Any group of exponent 2 is necessarily abelian ([DF91, page 23]) soG∗

= {1} is trivial.

Hence the map Σ → G is surjective, so M is spanned by im(ρ|Σ). It follows from

Nakayama’s lemma that M is also spanned by im(ρ|Σ).


Choose a modular form fχ ∈ S2(Γ0(N fB), εχ) with the following properties:

1. af (p) ∈ Q2(i);

2. af (p) = χB(p) a(p) for all p|p ∈ {3, 11, 19, 29, 31, 41, 61, 79, 89, 101, 109}.

Indeed, one does exist. (One uses Cremona’s Modular Symbol Algorithm to produce a

modular form level 800; see [Cre97] and [Cre92].) Consider the linear map ψ : M → Ogiven by ψ(a, b) = tr a− tr b. By 6.5.4, M is spanned by ρ(σp) for all σp ∈ Σ, so the

value of ψ is determined on these matrices. However, with this choice of modular

form fχ, we see that ψ = 0. In particular, tr ρ1 = tr ρ2, so it follows that ρf and

χB ⊗ ρE, ` have isomorphic semisimplifications. Hence EB is modular.

The author would like to thank Bill Stein for his help in finding the modular form

described above. More information about this form can be found in the appendix.

While we have found a modular form of level 800, it actually comes from a newform

of level 160. The difference is that the newform has a nonzero Fourier coefficient at

the prime above 5.

6.6 The Icosahedral Representation ρB

We have seen that the representation ρB comes from the mod 5 representation of the

elliptic curve EB and its twists. We make a few remarks about its properties.

Proposition 6.6.1. Let ρB be the complex icosahedral representation as in proposi-

tion 6.5.1.

1. ρB is isomorphic to the icosahedral representation studied in [Buh78].

2. This representation is the Galois deformation of a modular form of weight 2

and level 800 over Q. Explicitly, there is a modular form fχ ∈ S2(Γ0(800), εχ)

and a prime ideal λ ⊆ Q(√

5,√−1) lying above 5 such that

L(ρB, s) ≡ L(fχ, s) (mod λ) (6.36)


We choose our notation slightly different from that of [Buh78]. We replace j =

2 Re(ζ52) 7→ ε = 2 Re(ζ5) by choosing a different representation of the 5-cycles. We

also replace i 7→ −i via complex conjugation.

Recall that ω5 : 2 7→ i is our canonical definition. With this choice, the prime

ideal in the proposition is λ = (2− i, 2− ε).

Proof. Let ρ′B : GQ → GL2(C) be the map studied in [Buh78]. The field fixed

by the kernel of the projectivization is Q(qB) as studied at the beginning of the

chapter. Proposition 6.2.1 shows that, for any elliptic curve E defined over Q(√

5)

with j(E) = jB, we have Q(E[5]x) = Q(qB, ζ5). This is field fixed by the kernel of

projectivization of the mod 5 representation ρE,5. Hence, upon considering the base

change ρB, we find that Q(qB) is the field fixed by the kernel of its projectivization.

That is, ρB′ ∼= ρB since they have the same kernel. (Indeed, there are only two

irreducible projective complex representations of A5; they differ only at the 5-cycles.)

The fact that both representations have the same nebentype:

det ρ′B = (−1

·) = det ρB since

det ρB ≡ det ρχ,5 ⊗ χ5

≡ εχ ⊗ ω5 = εB mod 5(6.37)

(where χ5 is the cyclotomic character) shows that they must be isomorphic up to a

quadratic twist. To check that they are really the same, we use a modified proof as in

Theorem 6.5.1 to see that it suffices to check the traces for primes in Σ as in (6.35).

(This information can be found in the appendix.)

The mod 5 representation is ρE, 5∼= χ ⊗ ρB|G and the 5-adic representation is

ρE, 5∼= χ ⊗ ρχ, 5|G, where G = GQ(

√5). Following definition 5.2.2, it is clear that

E is an elliptic curve associated to ρB so by theorem 5.5.1 there is a prime ideal λ

lying above 5 such that L(ρB, s) ≡ L(ρχ, 5, s) (λ). Theorem 6.5.1 shows that we may

choose ρχ, 5 so that it is associated to a modular form, so the proposition follows.

Unfortunately, while we know that fχ and ρB are congruent to each other modulo

a prime ideal lying above 5, we have not been able to prove that ρB is modular. The

proof is this fact is the focus of [Buh78].


Most of the current results on Galois deformations — most notably Diamond’s

[Dia96] generalizations of the results in Wiles’ [Wil95] — assume that the abelian

variety Aχ associated with the modular form fχ ∈ S2(Γ0(800), εχ) is semistable.

As mentioned before, fχ seems to come from a newform of level 160, which is not

semistable, so a different approach will be needed.

Chapter 7

Final Remarks

We list a few remarks on how the methods outlined in this thesis can be used as an

attack for Artin’s Conjecture for representations of Platonic Type. (See definition

5.2.1.)

Consider the following definition.

Definition 7.0.1 (F -Curves). Let E be an elliptic curve defined over Qa, and F

be a number field. E is said to be an F -curve if it is isogenous over Qa to all of its

conjugates Eσ for σ ∈ GF . We say that E is a Q-curve if F = Q.

Proposition 7.0.2. Let ρ : GF → GL(V ) a representation of Platonic Type, and

assume that there exists an associated elliptic curve E ∈ Eρ which is an F -curve.

Then there exists a Hecke character χ such that χ ⊗ ρE, ` is isomorphic to the base

change of an `-adic representation ρχ, ` defined over F . Moreover, ρ is a deformation

of ρχ, `.

Proof. The notation below is the same as that in theorem 5.5.1.

Assume that E ∈ Eρ is an F -curve, and let χ be the corresponding Hecke character

as in theorem 5.5.1. Consider the twisted `-adic representation χ⊗ ρE, `. We wish to

show that it is Galois invariant. The idea here is to use the results of theorem 5.5.1

which shows that E is “almost” isogenous to a twist of its conjugate Eσ.

Denote a(p) and a(pσ) as the traces of Frobenius for E and Eσ, respectively. If

E and Eσ are isogenous over Qa, then E is isogenous to a quadratic twist of Eσ over

88

CHAPTER 7. FINAL REMARKS 89

F(j(E)

). Lemma 6.4.1 states that a(p) = Θσ(p) · a(pσ) for some quadratic character

Θσ. This gives

χ(pσ) a(pσ) ≡ ρ(N p) ≡ χ(p) a(p) ≡ Θσ(p)χ(p) a(pσ) (mod λ) (7.1)

so that χ(pσ) ≡ Θσ(p)χ(p). Theorem 5.5.1 states that χσ−1 is also a quadratic

character, so we must have equality. This shows that χ⊗ ρE, ` is Galois invariant, so

it must be the base change of some `-adic representation ρχ, ` defined over F . Clearly

ρ|G ≡ χ ⊗ ρE, ` = ρχ, `|G (λ) so the base change of ρ is a deformation of the base

change of ρχ, `. The proposition follows.

Conjecture 7.0.3. Let ρ : GF → GL(V ) a representation of Platonic Type. Then

there is an associated elliptic curve E ∈ Eρ which is an F -curve.

The main evidence for this conjecture comes from both 1) the example of EB

defined in (6.11) studied in the previous chapter, and 2) the result in theorem 5.5.1

which states that the traces tr ρE, ` are congruent to a quadratic twist of the traces

tr ρσE, ` for any associated curve E ∈ Eρ.

If this conjecture is true, then the proof of automorphicity of a representation ρ

should follow from the proof of automorphicity of an `-adic representation defined over

F . The associated elliptic curves E ∈ Eρ are defined over a biquadratic extension of

F , so the proof of automorphicity of an arbitrary associated curve E is more difficult.

Consider the case when F = Q. Recent work of Ken Ribet [Rib92], Ki-ichiro

Hashimoto [Has99], Yuji Hasagewa [Has97], Josep Gonzalez, and Joan Lario [GL98]

strongly suggest that a proof of the conjecture that all Q-curves are indeed modular

will soon be complete. We conjecture that Artin’s Conjecture for tetrahedral, octa-

hedral, and icosahedral representations over Q will indeed follow from deformations

of representations coming from biquadratic Q-curves.

Appendix A

Tables and Data

A.1 Introduction

Given the elliptic curve

EB : y2 = x3 + (5−√

5)x2 +√

5x

we have the associated quantities

jB = 86048− 38496√

5, fB = p62 p2

5; where ε =−1 +

√5

2.

The following tables are related to this curve. They were all generated on a Power

Macintosh running Mathematica r3.0. Total run time for all of these tables in this

appendix was about six months.

The author would like to thank University of California at Berkeley graduate stu-

dent William Stein for permission to use his emulation (HECKE) of John Cremona’s

Modular Symbol Algorithm [Cre97]. It was helpful in verifying the modularity of the

elliptic curve. The program can be found at the web site

http://shimura.math.berkeley.edu/~was/Tables/hecke.html

The author would also like to thank his advisor, Dan Bump, for usage of his

machine, a desktop running Linux.

90

APPENDIX A. TABLES AND DATA 91

A.2 Basic Properties of Q(√

5)

We make a few remarks here to explain how the entries in the following tables were

created.

The field Q(√

5) has narrow class number 1 with ring of integers Z[ε], which means

that every prime ideal p above a rational prime p may be identified with a totally

positive generator a+ b ε for some rational integers a and b. Recall that there are two

possibilities for primes p lying above p: either p is inert and p = (p), or p splits and

p = (a+ b ε) where a2 − a b− b2 = p is the norm.

Proposition A.2.1. Let p be a prime lying above a split rational prime p. Either

p or its Galois conjugate has a unique totally positive generator a + b ε such that

a > 3 b > 0 and a2 − a b− b2 = p.

This proposition allows us to identify primes lying above rational split primes

by solving a Diophantine equation subject to a simple inequality. This is helpful in

cutting down the search time for totally positive generators.

Proof. Consider the family

Fp =

{a+ b ε ∈ Z[ε]

∣∣∣∣ a2 − a b− b2 = p, a+ b ε is totally positive.

}Fix α(+) = a0 + b0 ε ∈ Fp such that b0 is positive. Its conjugate is α(−) =

(a0 − b0) − b0 ε, so a0 > b0 because α is assumed to be totally positive. Any other

element of Fp must be in the form ε2k α(±) by the unique factorization of Q(√

5). We

will use this fact to impose a condition on α so that it will be the “minimal” element

of Fp.

Set ak + bk ε = ε2k α(±). When k = −1, we have b−1 = ±(a0 + b0) so |b−1| =

a0 + b0 > b0. On the other hand, when k = 1, we have b1 = ±(2 b0 − a0). The

condition |b1| > b0 is equivalent to saying that either a0 > 3 b0 or a0 < b0. The latter

is impossible by the preceding paragraph so |b1| > b0 if a0 > 3 b0 > 0. Hence, there


does exist at least one a + b ε with a > 3 b > 0 and a2 − a b − b2 = p; simply choose

a+ b ε ∈ Fp with b > 0 smallest.

Conversely, it is clear that Fp has a unique a+ b ε with b > 0 minimal since every

subset of Z≥0 has a smallest element. The inequality a > 3 b > 0 forces a+ b ε to be

totally positive.

The following proposition shows how to effectively calculate quadratic symbols.

Proposition A.2.1. Let p be a prime lying above a rational prime p.

1. Let p 6= 2 be an inert prime. For all α ∈ Z[ε],(α

p

)≡ α

p2−12 mod p.

2. Let p be a split prime. For all α = c+ d ε ∈ Z[ε],(α

p

)=

(D

p

)where D = b (b c− a d), p = (a+ b ε) .

Proof. The statement when p lies above an inert prime is clear from the definition of

the quadratic residue symbol, so assume that it lies above a split prime. Choose a

generator a+ b ε for p, and consider the equation x2 ≡ c+ d ε (p). Upon multiplying

both sides by b2 we find that it is equivalent to the (b x)2 ≡ D (p). (Note that

a ≡ −b ε (p).) The result follows.


A.3 Structure of Gal(Q(√

5){2,5}/Q(√

5))

During the proof of Theorem 6.5.1 we exhibited the isomorphism

Gal(Q(√

5){2,5}/Q(√

5))∼= (Z2)

4 ; σp 7→{(

2

p

),

(2− ε

p

),

(−1

p

),

(ε

p

)}.

Recall that p2 = (2) and p5 = (2−ε) are the primes lying above 2 and 5, respectively.

The following table checks that the subset Σ ⊆ GQ(√

5) given by

Σ = {1} ∪{σp

∣∣ p lies above 3, 11, 19, 29, 31, 41, 61, 79, 89, 101, or 109}

is indeed surjective.


Table A.1: Structure of Gal(Q(√

5){2,5}/Q(√

5))

Rational Prime p Ideals p Above

(2

p

) (2− ε

p

) (−1

p

) (ε

p

)3 3 + − + −11 4 + ε − − − −

3− ε − − − +

19 5 + ε − + − −4− ε − + − +

29 6 + ε − − + +

5− ε − − + +

31 7 + 2 ε + − − −5− 2 ε + − − +

41 7 + ε + + + −6− ε + + + −

61 10 + 3 ε − + + −7− 3 ε − + + −

79 11 + 3 ε + + − +

8− 3 ε + + − −89 10 + ε + − + +

9− ε + − + +

101 13 + 4 ε − + + +

9− 4 ε − + + +

109 11 + ε − − + −10− ε − − + −


A.4 Group Structure of EB(Fp)

The following table contains information about the group structure for the curve

modulo prime ideals p above primes p less than 100. Recall that we have bad reduction

only at the primes lying above 2 and 5, with additive reduction in both cases.

Although the groups are expressed in a primary decomposition, it is well known

that EB(Fp) = Zn ⊕ Zm n for some integers m and n such that N p ≡ 1 (mod n).

Table A.2: Structure of EB modulo Primes

Rational Prime p Ideals p Above EB(Fp) Order of the Group

3 3 Z2 ⊕ Z4 8

7 7 Z2 ⊕ Z2 ⊕ Z11 44

11 4 + ε Z4 ⊕ Z3 12

3− ε Z2 ⊕ Z2 ⊕ Z3 12

13 13 Z2 ⊕ Z73 146

17 17 Z2 ⊕ Z137 274

19 5 + ε Z2 ⊕ Z4 ⊕ Z3 24

4− ε Z8 ⊕ Z3 24

23 23 Z2 ⊕ Z2 ⊕ Z3 ⊕ Z41 492

29 6 + ε Z2 ⊕ Z17 34

5− ε Z2 ⊕ Z13 26

31 7 + 2 ε Z4 ⊕ Z7 28

5− 2 ε Z2 ⊕ Z2 ⊕ Z9 36

37 37 Z2 ⊕ Z11 ⊕ Z59 1298

41 7 + ε Z2 ⊕ Z3 ⊕ Z7 42

6− ε Z2 ⊕ Z3 ⊕ Z7 42

43 43 Z2 ⊕ Z4 ⊕ Z233 1864

47 47 Z2 ⊕ Z2 ⊕ Z9 ⊕ Z59 2124

53 53 Z2 ⊕ Z3 ⊕ Z467 2802



59 9 + 2 ε Z8 ⊕ Z7 56

7− 2 ε Z2 ⊕ Z4 ⊕ Z7 56

61 10 + 3 ε Z2 ⊕ Z3 ⊕ Z11 66

7− 3 ε Z2 ⊕ Z29 58

67 67 Z2 ⊕ Z4 ⊕ Z7 ⊕ Z79 4424

71 9 + ε Z4 ⊕ Z3 ⊕ Z7 84

8− ε Z2 ⊕ Z2 ⊕ Z3 ⊕ Z5 60

73 73 Z2 ⊕ Z2729 5458

79 11 + 3 ε Z32 ⊕ Z3 96

8− 3 ε Z2 ⊕ Z32 64

83 83 Z2 ⊕ Z4 ⊕ Z881 7048

89 10 + ε Z2 ⊕ Z9 ⊕ Z5 90

9− ε Z2 ⊕ Z9 ⊕ Z5 90

97 97 Z2 ⊕ Z4793 9586


A.5 Supersingular Primes of EB

There are exactly four rational primes p less than 500 for which there is a prime ideal

p yielding a supersingular curve. The following table contains this information.

It seems quite likely that no inert primes are supersingular.

Table A.3: Supersingular Primes of EB


11 4 + ε Z4 ⊕ Z3 12

3− ε Z2 ⊕ Z2 ⊕ Z3

41 7 + ε Z2 ⊕ Z3 ⊕ Z7 42

6− ε Z2 ⊕ Z3 ⊕ Z7

89 10 + ε Z2 ⊕ Z9 ⊕ Z5 90

9− ε Z2 ⊕ Z9 ⊕ Z5

331 20 + 3 ε Z4 ⊕ Z83 332

17− 3 ε Z2 ⊕ Z2 ⊕ Z83


A.6 Coefficients of the L-Series

The following table contains information about the coefficients of the L-series attached

to the elliptic curve. Information for all rational primes p less than 360 is listed below.

The coefficient a(p) is given by the relation

a(p) = Np + 1− |EB(Fp)| = χB(p) ·

aχ(p) if p splits;

aχ(p)2 − 2 εχ(p) p if p is inert.

in terms of the Hecke character χB : Z[ε]→ C having the properties

χB(nσ) =

(−2

N n

)χB(n) and χB(n)2 = εχ(N n);

and the Fourier coefficients aχ(n) associated with the modular form

fχ(τ) =∑n≥1

aχ(n) qn ∈ S2

(Γ0(800), εχ

)of nebentype εχ =

(−1

·

)⊗ ω5.

(We fix ω5 as the Dirichlet character modulo 5 that maps 2 7→ i.) We also consider

the Fourier coefficients aB(n) associated to the modular form in [Buh78]:

fB(τ) =∑n≥1

aB(n) qn ∈ S1

(Γ0(800), εB

)of nebentype εB =

(−1

·

).

We choose our notation slightly different from that of [Buh78]. We replace j =

2 Re(ζ52) 7→ ε = 2 Re(ζ5) by choosing a different representation of the 5-cycles. We

also replace i 7→ −i via complex conjugation.

For all rational p we have the congruence aχ(p) ≡ aB(p) mod(2− i, 2− ε

).


Table A.4: Coefficients of the L-series attached to EB

Rational Prime p Ideals p Above a(p) χB(p) aχ(p) aB(p)

2 2 0 0 0 0

3 3 2 −i −2 (1− i) i

5 2− ε 0 0 0 0

7 7 6 −i −2 (1 + i) i ε

11 4 + ε 0 i 0 0

3− ε 0 i

13 13 24 −i 1 + i ε

17 17 16 −i −5 (1− i) 0

19 5 + ε −4 1 −4 i (1 + ε)

4− ε −4 1

23 23 38 i −2 (1− i) i

29 6 + ε −4 −i 4 i 1 + ε

5− ε 4 i

31 7 + 2 ε 4 −i −4 i i

5− 2 ε −4 i

37 37 72 i 1− i −1

41 7 + ε 0 1 0 0

6− ε 0 1

43 43 −14 −i 6 (1− i) i ε

47 47 86 −i 2 (1 + i) −i ε53 53 8 −i −7 (1 + i) −1

59 9 + 2 ε 4 −1 −4 i (1 + ε)

7− 2 ε 4 −1

61 10 + 3 ε −4 1 −4 1

7− 3 ε 4 −1

67 67 66 i 10 (1 + i) 0

71 9 + ε −12 −i 12 i i ε



8− ε 12 i

73 73 −128 i −3 (1 + i) 1

79 11 + 3 ε −16 1 −16 −i (1 + ε)

8− 3 ε 16 −1

83 83 −158 −i 2 (1− i) −i89 10 + ε 0 i 0 0

9− ε 0 i

97 97 −176 −i −3 (1− i) −ε101 13 + 4 ε −6 1 −6 −1

9− 4 ε 6 −1

103 103 134 i 6 (1− i) −i (1 + ε)

107 107 −142 i −6 (1 + i) i

109 11 + ε −10 i −10 i 0

10− ε 10 −i113 113 −64 i −9 (1 + i) 1 + ε

127 127 54 −i −10 (1 + i) 0

131 12 + ε −8 −i 8 i −i ε11− ε −8 −i

137 137 −272 −i 1− i −1

139 13 + 2 ε −12 −1 12 −i11− 2 ε −12 −1

149 15 + 4 ε −18 −i 18 i 1

11− 4 ε 18 i

151 16 + 5 ε −12 −i 12 i −i (1 + ε)

11− 5 ε 12 i

157 157 152 i −9 (1− i) −1

163 163 −318 −i −2 (1− i) i

167 167 326 −i 2 (1 + i) i (1 + ε)



173 173 8 −i 13 (1 + i) −1

179 17 + 5 ε 12 −1 −12 −i12− 5 ε 12 −1

181 14 + ε 10 −1 −10 0

13− ε −10 1

191 15 + 2 ε −20 i −20 i 0

13− 2 ε 20 −i193 193 −336 i −5 (1 + i) 0

197 197 344 i 5 (1− i) 0

199 16 + 3 ε 24 −1 −24 −i ε13− 3 ε −24 1

211 19 + 6 ε −16 −i 16 i i

13− 6 ε −16 −i223 223 246 i 10 (1− i) 0

227 227 −254 i −10 (1 + i) 0

229 17 + 3 ε −20 −i 20 i 0

14− 3 ε 20 i

233 233 −416 i −5 (1 + i) 0

239 16 + ε 8 −1 −8 i

15− ε −8 1

241 19 + 5 ε 16 −1 −16 −1

14− 5 ε 16 −1

251 17 + 2 ε −24 i −24 i i

15− 2 ε −24 i

257 257 −416 −i 7 (1− i) 1 + ε

263 263 454 i −6 (1− i) −i ε269 19 + 4 ε −10 −i 10 i 0

15− 4 ε 10 i



271 17 + ε 20 i 20 i 0

16− ε −20 −i277 277 392 i −9 (1− i) −1

281 22 + 7 ε −8 −1 8 1 + ε

15− 7 ε −8 −1

283 283 −494 −i −6 (1− i) −i ε293 293 536 −i −5 (1 + i) 0

307 307 −414 i 10 (1 + i) 0

311 21 + 5 ε −28 i −28 i i ε

16− 5 ε 28 −i313 313 −176 i 15 (1 + i) 0

317 317 392 i 11 (1− i) −1

331 20 + 3 ε 0 −i 0 0

17− 3 ε 0 −i337 337 384 −i −23 (1− i) −ε347 347 −46 i −18 (1 + i) i (1 + ε)

349 22 + 5 ε 20 −i −20 i 0

17− 5 ε −20 i

353 353 −544 i 9 (1 + i) ε

359 24 + 7 ε −16 −1 16 −i ε17− 7 ε 16 1

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ELLIPTIC CURVES AND ICOSAHEDRAL GALOIS REPRESENTATIONSegoins/notes/thesis.pdf · 2012. 7. 3. · L-series and its functional equation. In the 1930’s, Emil Artin [Art65] conjectured