New RAMANUJAN MODULAR FORMS AND THE KLEIN QUARTICiml.univ-mrs.fr/editions/biblio/files/lachaud_ramanujan.pdf · 2006. 5. 24. · friendly dedicated to Mikha¨ıl Tsfasman Abstract

RAMANUJAN MODULAR FORMSAND THE KLEIN QUARTIC

GILLES LACHAUD

friendly dedicated to Mikhäıl Tsfasman

Abstract. In one of his notebooks, Ramanujan gave some algebraic relations

between three theta functions of order 7. We describe the automorphic char-acter of a vector valued mapping made from these theta series. This provides

a systematic way to establish old and new identities on modular forms for

the congruence subgroup of level 7, above all the parametrization of the Kleinquartic. From an historic point of view, this shows that Ramanujan discovered

the main properties of this curve with his own means. As an application, we

introduce a L-series in four different ways, generating the number of points ofthe Klein quartic over finite fields. From this we deduce the structure of the

Jacobian of a suitable form of the Klein quartic over finite fields, and some

congruence properties on the number of its points.

Contents

1. Introduction 12. A representation of G 33. Ramanujan’s theta series of order 7 44. Hecke theta series 65. Parametrization of the Klein quartic 76. The rational curve X0(7) 87. The elliptic curve X0(49) 108. Hecke series and cusp forms 139. L-series 1510. Jacobians and reduction over finite fields 17Appendix A. Ramanujan’s formulas verbatim 20References 21

1. Introduction

In the complex projective space P2(C), the Klein quartic X is the curve of genus 3by the homogeneous equation

Φ4(x, y, z) = 0,

whereΦ4(x, y, z) = x3y + y3z + z3x.

We plan firstly to discuss on a specific modular parametrization of the Klein quarticby some theta constants a, b, c of order 7. We show that the main properties ofthese series are deduced from the modularity of a vector valued holomorphic map(Theorem 3.2). These theta series are written in a set of six formulas appearing in

Date: December 16, 2005.

1

2 Ramanujan Modular Forms and Klein Quartic

Ramanujan’s Notebooks [28, bottom of p. 300]; this passage is reproduced here asAppendix A for convenience).Felix Klein discovered in 1879 a modular parametrization of the eponym quartic[19], which reduces ultimately to an identity between the theta series a, b, c, leadingto an isomorphism from the modular curve X(7) to X. The work of Klein isexplained and cleared in the seminal report of Elkies [10]. This is precisely thisidentity which was obtained thirty years later by Ramanujan, who was most likelyunaware of the results of Klein.In spite of the fact that such a parametrization is essentially unique, it appearsunder many different forms, and we find worthwile to offer a unified presentationof the functions involved in this construction. In particular, most of the “thetaconstant identities” of order 7 can be ultimately seen as equations defining dominantmorphisms of the Klein quartic to its quotient curves. As we shall see, all thenecessary tools for a basic geometrical study of the Klein quartic have been providedby Ramanujan, with his own means.It would be interesting to describe in detail the history of these constructions; weshall content here with refering to the original contributions of Klein and Ramanu-jan, to the analysis of some of the aforementioned formulas of Ramanujan by Berndt& Zhang [3, 4] and Craig [6], and to the works of Farkas & Kra [11] and Liu [22],where the reader will find some of the formulas obtained here in another guise.Actually, the present work is related to a question of Mikhäıl Tsfasman: can oneexpress the modular form of level 49 and of weight 2 in terms of Dedekind’s etafunction? A tentative answer is given by the identity of Prop. 8.2. This identity,and some other appearing here, and involving Eisenstein series (Prop. 6.1 and 6.2)seem to be new (to be more careful, we did not found them in the literature).Finally, we would like to point out that the interest for the Klein quartic knew arenewal at the end of the last century, in view of its applications to coding theoryand cryptography, starting with the contibutions of Hansen [17], Rotillon & Thiong-Ly [32], Duursma [7], and Pellikaan [25]; see also [20].This article is organized as follows. The definition of Klein’s representation of thegroup G = PSL(2,F7) in a three-dimensional space is recalled in section 2. Thisrepresentation is needed in section 3 in order to state our first main Theorem 3.2,giving the automorphy properties of the set of three Ramanujan’s theta series a, b, c.These series have been rediscovered by Selberg, twenty years later. The transitionto three Hecke’s theta series x, y, z is performed in section 4. Theorem 3.2 impliesimmediately that x, y, z give a parametrization of the Klein quartic X, defining anisomorphism from X(7) onto X; this is done in section 5, followed by a discussionof various other proofs.The second part of the paper is devoted to subcoverings of the Klein curve, namely,the modular curve X0(7), and above all X0(49). In any case we need some infor-mation on the former, because it is a subcovering of the latter. The curve X0(7)is introduced in section 6, and since this curve is rational, any suitably invariantexpression of x, y, z can be expressed in terms of the hauptmodul of that curve. Thisgive rise to remarkable identities. Three of the aforementioned formulas of Ramanu-jan are of this kind: they lead to Klein’s relation giving the explicit formula for thecovering X(7) → P1 of degree 168. On the way, we obtain some identities involvingEisenstein series. Then a complete description of the morphisms connecting thethree curves is given in section 7.The third part of the paper is devoted to arithmetical applications. Since X0(49)is an elliptic modular curve, there is essentially only one cusp form of weight 2 forΓ0(49): we show in section 8 that this form can be expressed in terms of x, y, z, andthat it is in fact the theta series of a Hecke Grössencharakter. We establish in section

Ramanujan Modular Forms and Klein Quartic 3

9 a quadruple coincidence between four L-series, namely, on one side, the L-seriesof that modular cusp form (thus coming from the theory of automophic forms), andthree series on the other side: one associated to a Hecke Grössencharakter (comingfrom algebraic number theory), one associated to some Jacobi sums (coming fromthe theory of exponential sums), and eventually the L-series of the modular ellipticcurve of level 49 (coming from from algebraic geometry), a modest instance of theLanglands program. These identifications give in section 10 some useful informa-tions on the values of the Frobenius endomorphism of the reduction of the Kleinquartic over finite fields, and congruence properties of its number of points. We alsogive in that section the structure (in terms of Hermitian modules) of the Jacobianof the form of the Klein quartic admitting G as a group of automorphisms. Thiscurve has sometimes a maximal or minimal number of points.

2. A representation of G

Felix Klein discovered that the group of automorphisms of X is isomorphic to thesimple group G = PSL(2,F7) of order 168. One can be more precise. Recallthat the principal congruence subgroup of level 7 is the normal subgroup Γ(7) ofΓ(1) = PSL(2,Z) defined by the exact sequence

1 −−−−→ Γ(7) −−−−→ Γ(1) f−−−−→ G −−−−→ 1where f(γ) = γ(mod 7). Then, there is a representation

ρ : Γ(1) −−−−→ PGL(3,C)with kernel Γ(7) and leaving Φ4 invariant, defined as follows: if

T =[1 10 1

]and S =

[0 −11 0

]and if ζ = e2iπ/7, then

ρ(T ) =

ζ4 0 00 ζ2 00 0 ζ

and ρ(S) = −1√−7

ξ1 ξ2 ξ3ξ2 ξ3 ξ1ξ3 ξ1 ξ2

,with

ξ1 = ζ − ζ6 = 2i sin2π7, ξ2 = ζ2 − ζ5 = 2i sin

3π7, ξ3 = ζ4 − ζ3 = −2i sin

π

7,

and ρ(S) is a real symmetric orthogonal matrix.We should note in passing that P = (1 : 0 : 0) belongs to X, as well as

ρ(S)P = (sin2π7

: sin3π7

: − sin π7

),

and this implies the identity [3, p. 184]:

sin2 π7sin 3π7

−sin2 2π7sin π7

+sin2 3π7sin 2π7

= 0.

Recall that Γ(1) is defined by the presentation

< s, t; s2 = (st)3 = 1 >

and S and T satisfy these relations. Similarly, G is defined by the presentation

< s, t; s2 = t7 = (st)3 = (st3)4 = 1 >

satisfied by ρ(S) and ρ(T ). The group G contains a subgroup of order 3, generatedby the image of

H = ST−3STST−1ST 3 =[2 77 25

],


and

ρ(H) =

0 1 00 0 11 0 0

.The group Γ(1) operates by homographies on the half-plane H = {z | Im(z) > 0}and its extension H∗ obtained by including the cusps. The quotient Y (7) = Γ(7)\His a Riemann surface since Γ(7) is torsion free, and the modular curve of level 7 isthe compact Riemann surface X(7) = Γ(7)\H∗. This Riemann surface has genus3 and G acts faithfully on it by construction. From this, one deduces that X(7) isisomorphic to the Klein quartic X, which is, up to isomorphism, the only Riemannsurface of genus 3 on which G acts faithfully [10, p. 70].

3. Ramanujan’s theta series of order 7

In order to define an explicit isomorphism from X(7) to X, we start from threetheta series: if q = exp(2iπz) and z ∈ H, we define

a(z) = −∑n∈Z

(−1)n q(14n+5)2/56,

b(z) =∑n∈Z

(−1)n q(14n+3)2/56,

c(z) =∑n∈Z

(−1)n q(14n+1)2/56,

and we define an holomorphic map of H into C3:

A(z) =

a(z)b(z)c(z)

.These series are those of Ramanujan’s definition (D) of the Appendix. We havemade a slight change by setting (a, b, c) = (−w, v, u).

3.1. Proposition. If z ∈ H, the following relations hold:A(z + 1) = e−iπ/4 ρ(T )A(z),

A(−1/z) = e3iπ/4√z ρ(S)A(z),

where 0 < arg√z ≤ π/2.

Proof. The theta constants are the functions

θ

[εε′

](z) = eiπεε

′/2∑n∈Z

eiπnε′q(n+

ε2 )

2/2

where ε and ε′ are two real numbers [11, p. 72 sqq]. If k and l are integers, then

θ

[l/k1

](kz) = eiπl/2k

∑n≡l(mod 2k)

(−1)nqn2/8k.

If k is odd, the modified theta constants of order k are

ϕl(z) = θ[(2l + 1)/k

1

](kz), l = 0, . . . ,

k − 32

.

One checks that the functions a, b, c are proportional to the three modified thetaconstants of order 7:

a(z) = −e−5iπ/14 ϕ2(z), b(z) = e−3iπ/14 ϕ1(z), c(z) = e−iπ/14 ϕ0(z).The required relations then follow from the transformation formulas of modifiedtheta constants [11, p. 217]. �


We define now

j(γ, z) = cz + d if z ∈ H and γ =[a bc d

]∈ Γ(1).

Recall that if

η(z) = q1/24∞∏n=1

(1− qn)

is the Dedekind eta function, if γ ∈ Γ(1), and if z ∈ H, then [11, p. 339]

η(γz) = vη(γ) j(γ, z)1/2 η(z),

with a multiplier vη(γ) independent of z which is a 24-th root of unity, and wherethe argument of the square root is in [0, π[.

3.2. Theorem. If z ∈ H, then

A(γz) = vη(γ)−3j(γ, z)1/2 ρ(γ)A(z) if γ ∈ Γ(1).

Proof. Let X1(z) = η3(z)A(z). It is sufficient to prove that

X1(γz) = (cz + d)2 ρ(γ)X1(z) if γ ∈ Γ(1),

and for this it suffices to establish that

X1(z + 1) = ρ(T )X1(z),X1(−1/z) = z2 ρ(S)X1(z).

But this is a consequence of Proposition 3.1, the formulas

η(z + 1)3 = eiπ/4 η(z)3, η(−1/z)3 = e−3iπ/4 z3/2 η(z)3,

being taken into account. �

Since Γ(7) = ker ρ, we deduce from Theorem 3.2 that

A(γz) = vη(γ)−3j(γ, z)1/2 A(z) if γ ∈ Γ(7),

This means that the functions a, b, c are automorphic forms of weight 1/2 for Γ(7)with the same factor of automorphy. In order to describe the product expansion ofa, b, c, we use the standard notation

(a; q) =∞∏n=0

(1− aqn),

if a ∈ C× and if |q| < 1, in such a way that η(z) = q1/24(q; q). From the productformula [11, p. 141] we deduce

θ

[l/k1

](kz) = exp(iπl/2k) q(l

2/8k) (q(k−l)/2; qk)(q(k+l)/2; qk)(qk; qk),

and the expansions are

a(z) = − q25/56 (q; q7) (q6; q7) (q7; q7),b(z) = q9/56 (q2; q7) (q5; q7) (q7; q7),c(z) = q1/56 (q3; q7) (q4; q7) (q7; q7).

This implies

(1) a(z)b(z)c(z) = −η(z) η(7z)2,

a relation equivalent to Ramanujan’s formula (F).


3.3. Remark. If a ∈ C× and if |q| < 1, we use the notations (a; q)0 = 1, and

(a; q)n =n−1∏m=0

(1− aqm) (n ≥ 1).

In two articles, [30] and [31], Selberg rediscovered the series a, b, c, found the aboveproduct expansions, and established the identities

a(z) =∞∑n=0

(q2; q2)∞(q2; q2)n

q2n(n+1)

(−q;−q)2n+1,

b(z) =∞∑n=0

(q2; q2)∞(q2; q2)n

q2n(n+1)

(−q;−q)2n,

c(z) =∞∑n=0

(q2; q2)∞(q2; q2)n

q2n2

(−q;−q)2n.

He also set up a link between a, b, c and Ramanujan’s seventh order mock thetafunctions [29, pp. 127–131].

If Γ is a congruence subgroup of Γ(1), and if k is a positive integer, we denote byMk(Γ) the space of cusp forms of weight k for Γ,i.e. satisfying

f(γz) = j(γ, z)k f(z) if γ ∈ Γ,

and by Sk(Γ) the subspace of cusp forms. The above product expansions show thata, b, c vanish at the infinite cusp of X(7), and Theorem 3.2 implies that they vanishat any other cusp, since the operation of Γ(1) on the cusps is transitive. Since q1/7

is a local parameter for X(7) at the infinite cusp, the eighth powers of a, b, c arecusp forms and belong to S4(Γ(7)).

4. Hecke theta series

The ring of integers of the imaginary quadratic field K = Q(√−7) is Z[α], where

α =−1 +

√7

2.

Let f = (√−7) be the unique prime ideal above 7 in Z[α]. We define three Hecke

theta series, where z ∈ H and ξ ∈ Z[α]:

x(z) = −∑

ξ≡1(mod f)

ξ qξξ̄/7,

y(z) =∑

ξ≡2(mod f)

ξ qξξ̄/7,

z(z) =∑

ξ≡4(mod f)

ξ qξξ̄/7,

and the holomorphic map of H into C3:

X(z) =

x(z)y(z)z(z)

4.1. Proposition. The functions x, y, z form a basis of the space S2(Γ(7)), and

X(z) = η3(z)A(z).


Proof. If ξ = x+ yα, then

ξξ̄ = Q(x, y), where Q(x, y) = x2 − xy + 2y2.

Now it follows from Schoeneberg’s theorem [24, p. IV-22] that for any non constantaffine function L(x, y), the series∑

(x,y)∈Z2L(x, y) e2iπzQ(x,y)

belongs to S2(Γ(7)). By taking appropriate affine functions, one deduces from thisthat x, y, z belongs to S2(Γ(7)). Hence, the coordinates of X − η3A belongs alsoto S2(Γ(7)) ; by checking q-expansions, one ensures that these cusp forms vanishidentically. The q-expansions show as well the linear independence of these forms,and the dimension of S2(Γ(7)) is equal to the genus of X(7), which is 3. �

From Theorem 3.2 we deduce:

4.2. Corollary. If z ∈ H, then

X(γz) = j(γ, z)2 ρ(γ)X(z) if γ ∈ Γ(1). �

5. Parametrization of the Klein quartic

The first main result is that the cusp forms x, y, z leads to a parametrization of theKlein quartic:

5.1. Proposition. The cusp forms x, y, z satisfy

(2) Φ4(x, y, z) = 0,

and the mapX(7) −−−−→ X

deduced from X by projection is an isomorphism.

Observe that the two holomorphic mappings from X(7) to P2 deduced from X andA by projection are identical, i.e. (x : y : z) = (a : b : c).

Proof. As in [10]. Corollary 4.2 implies that Φ4(x, y, z) is a cusp form of weight8 for Γ(1), but the only such form is zero. The linear map f(z) 7→ f(z)dz is anisomorphism from S2(Γ(7)) to the space H0(X(7),Ω1) of holomorphic differentialforms on X(7), hence, the map X is the canonical embedding, which is an isomor-phism since X(7) is of genus 3 and not hyperelliptic (recall that we know that X(7)is isomorphic to the Klein quartic). �

The identity (2) in terms of (a, b, c) is equivalent to Ramanujan’s formula (E). Werecord here some other proofs.(i) The classical Jacobi theta series is

ϑ1(x, z) = −θ[11

](x, z) = −i

∑n∈Z

(−1)neiπ(2n+1)xq(2n+1)2/8

where we used the notations of [11] in the middle term. The expression of A interms of this theta series is

tA(z) = [eiπz/7 ϑ1(z, 7z),−e4iπz/7 ϑ1(2z, 7z),−e9iπz/7 ϑ1(3z, 7z)].

Proofs of (2) using these notations appears in [3, p. 176] and [6]. We remark inpassing that the functions x, y, z can be fully expressed in terms of theta functions,since [11, p. 289]:

ϑ′1(0, z) = 2πη3(z).


(ii) If one applies the Fricke involution w7z = −1/7z and makes use of the trans-formation formula of ϑ1(x, z), we obtain:

tA(w7z) = e−iπ/4√z [−ϑ1(1/7, z), ϑ1(2/7, z), ϑ1(3/7, z)].

Then (2) is equivalent to the same identity applied to the three functions inside thebrackets. A proof of this relation, based on elliptic functions, is given in [22].(iii) Proposition 5.1 is proven in [11, p. 246] by an analysis of the divisors of a, b, cand the identity (2) for these functions is proved again in [11, p. 256] as a particularcase of a general three term identity.(iv) An approach of (2) based on infinite determinants is to be found in [8].(v) The morphism of Proposition 5.1 is expressed in terms of Klein forms in [16].

6. The rational curve X0(7)

Recall that the Hecke modular curve of level N is the quotient X0(N) = Γ0(N)\H∗of the extended upper half-space by the congruence subgroup

Γ0(N) ={[a bc d

]∈ SL(2,Z) | c ≡ 0 (modN)

}(N ≥ 1).

Then Γ(7) is a normal subgroup of Γ0(7), and Γ0(7) is generated by Γ(7) andthe group generated by H and T . The quotient group Γ0(7)/Γ(7) is of order 21,isomorphic to the subgroup g of G made of upper triangular matrices. The naturalprojection defines a Galois covering

X(7) −−−−→ X0(7)with Galois group g. Recall also that

j7(z) = (η(z)/η(7z))4

is the hauptmodul of X0(7), that is, the unique univalent modular function for Γ0(7)holomorphic on H with a simple pole at z = i∞ and a simple zero at z = 0. Let

Φ3(X,Y, Z) = XY Z, Φ6(X,Y, Z) = XY 5 + Y Z5 + ZX5 − 5X2Y 2Z2.Then

Φ3(x, y, z) = −η(z)10η(7z)2, Φ6(x, y, z) = ∆(z) = η24(z).The first relation comes from (1) ; for the second, remark that Φ6(x, y, z) belongs tothe one dimensional space M12(Γ(1)) since Φ6 is G-invariant [10, p. 57]. We obtain

(3)Φ6(x, y, z)Φ3(x, y, z)2

= j7,

and we get a commutative diagram

X(7) −−−−→ X0(7)

∼yX ∼yj7X

Φ6/Φ3−−−−→ P1

The equality (3) means that

(4)1

xyz(x4

y+

y4

z+

z4

x) = j7 + 5,

a relation equivalent to Ramanujan’s formula (C). Such an identity is not due tochance: the function fieldM0(Γ0(7)) is the subfield ofM0(Γ(7)) left fixed by the Ga-lois group g. This means that if F (X,Y, Z) is any rational function, homogeneousof degree zero, invariant under cyclic permutation, and such that

F (ζ4X, ζ2Y, ζZ) = F (X,Y, Z),


then F (x, y, z) is a rational function of j7, and conversely. More generally, if we onlyassume that F is degree k, then F (x, y, z) is a meromorphic automorphic form ofweight 2k. As a first example, let

Φ9(X,Y, Z) = X5Y 4 + Y 5Z4 + Z5X4.

SinceΦ3Φ6 + 8Φ33 = −Φ9 modΦ4,

we obtain−xyz ( y

x4+

z

y4+

x

z4) = j7 + 8,

a relation equivalent to Ramanujan’s formula (B). A second example is given byKlein’s expression of the absolute invariant j in terms of j7, which can be obtainedas follows from Ramanujan’s formula (G). Let

Φ5(X,Y, Z) = −(X2Y 3 + Y 2Z3 + Z2X3),and

P (U, V ) = U2 + 13UV + 49V 2.Then the relation

(5) Φ35 = Φ3P (Φ6,Φ23) modΦ4

implies

xyz

(x

y2+

y

z2+

z

x2

)3= j27 + 13j7 + 49,

which is equivalent to (G). Let Φ14 be the G-invariant form of degree 14 obtainedby Klein as a differential determinant [10, p. 58], and

Q(U, V ) = U2 + 5.72 UV + 74 V 2.

Then

(6) Φ14Φ3 ≡ Φ5Q(Φ6,Φ23) modΦ4.From (5) and (6) we deduce

(7) Φ314Φ23 ≡ P (Φ6,Φ23)Q(Φ6,Φ23)3 modΦ4.

Denote by E4 the classical Eisenstein series of weight 4 for Γ(1). Then [10, p. 85]

E4 =Φ14(x, y, z)Φ6(x, y, z)2

.

Since j = E34/∆, we have

(8) j =Φ14(x, y, z)3

Φ6(x, y, z)7,

and dividing both sides of (7) by Φ23Φ76, we obtain Klein’s relation

(9) j = f7(j7), f7(x) = P (x, 1)Q(x, 1)3/x7.

Let Mk(Γ0(7), χ7) be the space of modular forms of type (k, χ7), with the characterχ7(d) = (−7/d): a form in this space satisfies

f(γz) = χ7(d) (cz + d)k f(z), γ =[a bc d

]∈ Γ0(7).

The function

k(z) =η(z)7

η(7z)belongs to M3(Γ0(7), χ7), and xyz = −k2 t, where

t(z) = (η(7z)/η(z))4 = 1/j7(z).


It is an easy exercise to see that Klein’s relation (9) is equivalent to two classicalidentities in the “Lost Notebook” of Ramanujan [26], which are

E34 = k4R(1, t), E6 = k2S(1, t),

where E6 is the Eisenstein series of weight 6 for Γ(1), and

S(U, V ) = U4 − 2.5.72 U3V − 32.74 U2V 2 − 2.76 UV 3 − 77 V 4.

It is worthwile to notice that these identities show that E4 and E6 are given by analgebraic expression involving only η(z) and η(7z). Note also that we shall give inProp. 6.1 a related identity expressing rationally E4 itself in terms of x, y, z and t.The series

Θ0(z) =∑ξ∈Z[α]

qξξ̄

belongs to M1(Γ0(7), χ7) by Schoeneberg’s theorem. Since

R(n) ={(x, y) ∈ Z2 | x2 + xy + 2y2 = n

}= 2

∑d|n

χ7(d),

we have

Θ0(z) = 1 +∞∑n=1

R(n)qn = 1 + 2∞∑n=1

(n

7)

qn

1− qn.

6.1. Proposition. We have the double equality

− xyz(

x

y2+

y

z2+

z

x2

)= Θ0k =

j27j27 + 5.72 j7 + 74

E4.

Hence, Θ is given by an algebraic expression involving only η(z) and η(7z).

Proof. From (6) we deduce that the first term is equal to the last one. In orderto prove the first equality, observe that since k ∈ M3(Γ0(7), χ7), both sides are inM4(Γ0(7)); check the q-expansions. �

Now Θ20 ∈M2(Γ0(7)). Since this space is one dimensional, with basis the Eisensteinseries

E(7)2 (z) = q

d

dqlog t(q) = 1 + 4

∞∑n=1

σ1(n)(qn − 7q7n),

we have actually E(7)2 = Θ20, another identity of Ramanujan [2, Entry 5(i),p. 467].

Moreover

6.2. Proposition. We have

x7 + y7 + z7

(xyz)2= E(7)2 .

Proof. The left member belongs to M2(Γ0(7)), with constant term 1. �

7. The elliptic curve X0(49)

Recall that there is, up to isomorphism, only one elliptic curve E defined over Q withcomplex multiplication by Z[α]. The absolute invariant is j(E) = j(α) = −153. AWeierstrass equation of such a curve is

(10) y2 = x3 − 35x+ 98.

The endomorphism of degree 2 providing the multiplication by α is

(x, y) 7→(α−2(x− 7(1− α)

2

x− α− 4), α−3y(1 +

7(1− α)2

(x− α− 4)2)).


The curve E has conductor 49, and is listed as the curve A(7) by Gross [14]. Anothercharacterization of this curve is given in Remark 7.3 below.A morphism from the Klein quartic X to E is defined as follows. Let

(11)

f =x

z+y

x+z

y=x2y + y2z + z2x

xyz,

g =x

y+y

z+z

x=xy2 + yz2 + zx2

xyz.

These expressions are invariant under cyclic permutations of the letters and definetwo functions on X, with poles located at the points (0 : 0 : 1), (0 : 1 : 0), (1 : 0 : 0),of respective orders 2 for f and 3 for g. By looking at the local expansions of thesefunctions at these poles, one finds for these functions the relation

(12) g2 − 3fg + g + f3 − 2f + 3 = 0,which defines a curve isomorphic to E (the absolute invariant is the same), withdiscriminant ∆ = −73. Namely, if

(13) f = −1− 1u, g = −1

2(3u

+v

u2)− 2 ,

then (u, v) belongs to the projective affine curve E with equation

(14) v2 = 28u3 + 21u2 + 4u,

which is a third model of E, since we recover (10) from (14) by the change ofvariables

u =x− 728

, v =y

28.

Hence, the couple (u, v) defines a dominant morphism

ϕ : X −−−−→ Ewhich identifies E with the quotient of X by the group of order 3 of cyclic permu-tations on three letters.This morphism has a modular counterpart. If f(z) is a modular form of weight kfor Γ0(N2), then α∗f(z) = f(z/N) is a modular form of the same weight for thesubgroup

Γ[(N) ={[a bc d

]∈ SL(2,Z) | b ≡ c ≡ 0(modN)

},

and conversely. Since Γ(N) ⊂ Γ[(N), we get a dominant morphism

X(N) α−−−−→ X0(N2).If N = 7, then the curve X0(49) is of genus one, and is isomorphic to E. Theconstruction of this isomorphism is obtained as follows. Firstly, we note that Γ[(7)is generated by Γ(7) and the group generated by H, whose elements performs cyclicpermutations of x, y, z. Then, substituting the forms x, y, z to the variables x, y, zin the relations (11) yields three forms which are modular for Γ[(7), because theyare modular for Γ(7) and for the group generated by H. Now, If f(z) is a modularform for Γ[(7), then f7(z) = f(7z) is modular for Γ0(49). Hence,

(x2y + y2z + z2x)7, (xy2 + yz2 + zx2)7, (xyz)7are three cusp forms of weight 6 for Γ0(49), and the two expressions

f = (x

z+

y

x+

z

y)7 , g = (

x

y+

y

z+

z

x)7 ,

define modular functions on X0(49). Then the functions u and v defined by

(15) f = −1− 1u, g = −1

2(3u

+v

u2)− 2 ,


satisfy (14), from which we deduce:

7.1. Proposition. The map

G : X0(49) −−−−→ E

defined by G(z) = (u(z), v(z)) is an isomorphism. �

This is the simplest case of modular theory, which ensures that if E is an ellipticcurve defined over Q of conductor N , there exists a surjective map X0(N) −→ E.There is a unique such map of minimal degree (up to composing with automor-phisms of E), and its degree is the modular degree of E. Here Prop. 7.1 showsthat E is of modular degree one. The following diagram commutes:

X(7) α−−−−→ X0(49)

∼yX ∼yGX

ϕ−−−−→ EWe now express u and v in terms of η(z), and for this purpose we use two functionsdefined by Fricke [12, p. 400-403]. The first one is t(z) = (η(7z)/η(z))4, and thesecond one is η(49z)/η(z). By comparing q-expansions, one proves that

u(z) = η(49z)/η(z).

Then t and u generates the function field of X0(49): the comparison of q-expansionsleads to

(16) v =2t− 7u(1 + 5u + 7u2)

1 + 7u + 7u2.

The two preceding identities, together with (15), give an expression of f and g interms of the eta function ; considering the relation

v

u2=

√4u3

+21u3

+28u,

we see that the second formula of (15) is equivalent to Ramanujan’s formula (A),and that the first one is equivalent to a formula of the Notebooks [2, Entry 17(v),p. 303].

7.2. Remark. The Fricke involution w49z = −1/(49z) is an automorphism ofX0(49)permuting the two cusps, and

t(w49z) =t(z)

49 u(z)4, u(w49z) =

17u(z)

, v(w49z) =v(z)

7 u(z)2,

hence, w49 corresponds to the involutive automorphism P 7→ P0 − P of E, whereP0 = (0, 0), given by

(u, v) 7→ ( 17u,v

7u2).

The expression (16) of v leads to

(17) 2t = 7u(1 + 5u + 7u2) + (1 + 7u + 7u2)√

28u3 + 21u2 + 4u,

which is also equivalent to a formula of the Notebooks [2, Entry 18(vi), p. 306].This equation was first proved by Watson in [39], and is called the modular relationfor the prime 7 [11, p. 392 and 399]. If we write (17) as t = ψ(u, v), we get


j7(z) = 1/ψ ◦ G(z), and the various diagrams combine themselves in order to givethe full picture

X(7) 3−−−−→ X0(49)7−−−−→ X0(7)

8−−−−→ X(1)

∼yX ∼yG ∼yj7 ∼yjX

ϕ−−−−→ E 1/ψ−−−−→ P1 f7−−−−→ P1

where the degrees of the coverings are indicated on the top horizontal arrows. Thisgives an explicit factorization of the “Equation of degree 168” of Klein, that is, thecovering of degree 168 of the projective plane by the Klein quartic given by (8).

7.3. Remark. Let E0 be an ordinary elliptic curve over Fp, together with its p-thpower Frobenius endomorphism F0 ∈ EndFp(E0), defined over Fp. Up to isomor-phism there is a unique elliptic curve E over Zp such that E0 is the reduction of Emodulo p and such that the map

EndZp(E) → EndFp(E0)is an isomorphism [33]. The curve E is the canonical lift of E0; in particular, thereis a unique endomorphism F ∈ EndZp(E) mapping to F0. Assume now that E isour elliptic curve with CM by Z[α]. In the equation of E given by (12), let f = 1−xand g = 1− 2x− y; we get

y2 + xy = x3 − x2 − 2x− 1,which reduces modulo 2 as the equation of an ordinary elliptic curve E0 over F2:

y2 + xy = x3 + x2 + 1.

Since the roots of the Frobenius automorphism F of E0 are −α and −ᾱ, we haveF 2 − F + 2 = 0,

hence, the ring EndFp(E0) is isomorphic to Z[α] and E is the canonical lift of E0.

8. Hecke series and cusp forms

The inclusion Z −→ Z[α] induces a ring isomorphism

Z/7Z ∼−→ Z[α]/(f),for instance α ≡ 3(mod f). Composing the inverse isomorphism with the quadraticresidue symbol gives a character on Z[α], given, for ξ ∈ Z[α], by

(ξ

f)2 =

1 if ξ is a non-zero square (mod f),−1 if ξ is a non-square (mod f),0 if ξ ≡ 0 (mod f).We denote by I(f) the group of the fractional non-zero ideals in K whose factoriza-tion in prime ideals does not involve f. If ξ ∈ K×, define

λ(ξ) = (ξ

f)2 ξ.

This is an even function which depends only of the ideal generated by ξ. Thus, ifa ∈ I(f) is generated by ξ, we set λ(a) = λ(ξ). The map

λ : I(f) −−−−→ K×

is a Hecke Grössencharakter of K with conductor f, according to the usual criterion[41]. We associate to this Grössencharakter its Theta series

Θ(z, λ) =∑

a∈I(f)

λ(a) qN(a)


where N(a) is the norm of the ideal a. We frequently write Θ(z, λ) = Θ(z) forsimplicity. This Theta series has a q-expansion

Θ(z) =∞∑n=1

cnqn

whose first terms are

Θ(z) = q + q2 − q4 − 3q8 − 3q9 + 4q11 − q16 − 3q18

+ 4q22 + 8q23 − 5q25 + 2q29 + 5q32 + 3q36 − 6q37 +O(q43).

8.1. Proposition. The cusp forms x, y, z satisfy

(x + y + z)7 = Θ,

and this function is the unique form of the one dimensional space S2(Γ0(49)) suchthat Θ(q) ∼ q if q → 0. Moreover

Θ(w49z) = −49 z2 Θ(z).

Proof. The Grössencharakter λ is an isomorphism of I(f) to the subgroup of K×

made of squares modulo f, hence

Θ(z) =12

∑ξ∈Z[α]

ξ qξξ̄ =∑

ξ≡1,2,4(mod f)

ξ qξξ̄,

and this implies the first formula. Now we know that (x + y + z)7 belongs to thespace S2(Γ0(49)), which is one-dimensional, since X0(49) is an elliptic curve. Sincethe sum of the columns of ρ(S) is equal to −1, we deduce from Cor. 4.2 that themodular form s1(z) = x(z) + y(z) + z(z) satisfies

s1(−1/z) = −z2 s1(z),from wich the second formula follows. �

One can express Θ in terms of classical modular forms on Γ0(49). A standardmodular form of weight 2 for Γ0(49) is the Eisenstein series

E(49)2 (q) = q

d

dqlog

η(q49)η(q)

= 2 +∞∑n=1

σ1(n)(qn − 49q49n),

where σ1(n) is the sum of divisors of n. Now Θ20(7z) is a modular form of weight2 for Γ0(49), and it is easy to see that 4E

(49)2 (z) = E

(7)2 (z) + 7E

(7)2 (7z), hence

(18) 4E(49)2 (z) = Θ0(z)2 + 7Θ0(7z)2.

8.2. Proposition. We haveΘ =

u

vE

(49)2 .

Hence, Θ is given by an algebraic expression involving only η(z), η(7z), and η(49z).

Proof. The normalized invariant differential form on the elliptic curve E is

ω =du

v,

from which one deduces thatΘ = v−1 q

du

dq,

and sinceE

(49)2 (q) = q

d

dqlog u(q) = u−1 q

du

dq,

one gets the required formula. �


9. L-series

The L-series of λ is defined if Re(s) > 1 by

L(λ, s) =∑

a∈I(f)

λ(a) N(a)−s =∏p6=f

(1− λ(p) N(p)−s)−1.

This is as well the L-series of the modular form Θ:

(19) L(λ, s) = L(Θ, s) = (2π)sΓ(s)−1∫ ∞

0

Θ(iy, λ)ys−1dy,

where Γ(s) is Euler’s Gamma function. From the relation Θ(w49z) = −49 z2 Θ(z),one deduces the functional equation

72−sL̂(λ, 2− s) = 7s L̂(λ, s), where L̂(λ, s) = (2π)−sΓ(s)L(λ, s).

Now let p be an odd prime number 6= 7. If (p/7) = +1 then (p) = pp̄ with a primep of K, thus N(p) = p, and there is a unique integer πp generating p such that

πp = ap + bp√−7, a2p + 7b2p = p, (

ap7

) = 1, bp 6= 0,

with ap and bp in Z, because p is odd; note that ap = Reπp = Reπp̄ depends onlyon p. In that case

λ(p) = πp.

If p = 2, we take p = (α) since αᾱ = 2, and we define πp = −ᾱ. If (p/7) = −1 thenp = p is inert in K, thus N(p) = p2, and

λ(p) = −p.

9.1. Remark. Let p be a prime number with p ≡ 1 (mod 7). Then (p/7) = +1 and(p) = pp̄. Let q be a prime ideal of L = Q(ζ7) above p. For every x ∈ Z[ζ7], primeto q, the 7-th power residue symbol (x/q)7 is the unique 7-th root of unity satisfyingthe condition

x(p−1)/7 ≡ (xq)7 (mod q).

The 7-th power residue symbol induces a multiplicative character χq of order 7 ofthe field κ = Z[ζ]/q with 7 elements. We are now able to define the Jacobi sum

J (q) = −∑x∈κ

χq(x(1− x)2).

Recall the following results [20]: the Stickelberger congruence Relation implies that(Jq) = p, hence, the value of these sums depends only on p, and we denote theircommon value by Jp. Then the Ihara congruence Relation implies

J (p) = λ(p),

in other words, the Grössencharakter λ is defined by the Jacobi sum Jp, a particularcase of Weil’s Theorem [40]. Thus the L-series of this family of exponential sums,namely

L(J , s) =∏p6=f

(1− J (p) N(p)−s)−1,

and the L-series of the Grössencharakter are identical:

L(J , s) = L(λ, s).

Since

L(λ, s) =∑

a∈I(f)

λ(a) N(a)−s =∞∑n=1

cnn−s,


one has

cp = Trλ(p) = Trπp = 2ap if (p/7) = +1,= 0 if (p/7) = −1.

From the product expansion we deduce that

L(λ, s) =∏p6=7

(1− cpp−s + p1−2s)−1.

9.2. Remark. The Chebotarev density theorem [21, Ch. VIII] tells us that the setS of primes p such that (p/7) = −1 has density 1/2. Since p ∈ S if and only ifcp = 0, the set of integers n ∈ N such that cn 6= 0 has density 0 [34, Th. 13].

The elliptic curve E has good reduction for any prime p 6= 7. If p is such a prime,we denote by E/Fp its reduction modulo p, and by P (E/Fp, T ) the numerator ofthe zeta function of E/Fp. The global L-series of E is then

L(E/Q, s) =∏p6=7

P (E/Fp, p−s)−1.

9.3. Proposition. If p 6= 7, then

P (E/Fp, T ) = 1− cpT + pT 2, and |E(Fp)| = p+ 1− cp.

MoreoverL(E/Q, s) = L(λ, s).

Proof. The two assertions are equivalent. Since E = X0(49), Shimura’s theorem[38, p.182] implies that L(E, s) is obtained from the Mellin transform of the uniquenormalized form in S2(Γ0(49)):

(20) (2π)−s Γ(s)L(E, s) =∫ ∞

0

Θ(iy)ys−1dy ;

now compare with (19). �

9.4. Remarks. (i) A direct proof of Proposition 9.3 can be obtained following themethod for CM elliptic curves explained in [37, Ch. II, § 10]. This method isapplied to the particular elliptic curve E in [14] and [27].(ii) Since cp = 0 if and only if (p/7) = −1, the set S is the set of primes for whichE/Fp is supersingular by Proposition 9.3.(iii) As observed by Shimura [38, p. 221], Formula (20) implies that L(E, 1) 6= 0.In fact, according to Gross and Zagier [15], one has

L(1, χ) =1

4π√

7Γ(

17)Γ(

27)Γ(

47).

This fact is in agreement with the conjecture of Birch and Swinnerton-Dyer, sinceE(Q) has only two points: (0,0) and the point at infinity [10, p. 73].

The following result provides recurring relations for cpn and |E(Fpn)|.

9.5. Corollary. let Tn be the nth Hecke operator. If (n, 7) = 1, then TnΘ = cnΘ.Moreover, if n ≥ 2, and if p 6= 7, then

cpn = cpn−1cp − pcpn−2 ,

and|E(Fpn)| = pn + 1− cpn + pcpn−2 .


Proof. Since

L(Θ, s) =∞∑n=1

cnn−s =

∏p6=7

(1− cpp−s + p1−2s)−1,

the series Θ is a common eigenvalue of the Hecke operators [38, Th. 3.43]. The re-currence relation for cpn is a consequence of the basic properties of Hecke operators,and this relation implies

1P (E/Fp, T )

=1

1− cpT + pT 2=

∞∑n=0

cpnTn,

from which one deduces

−P′

P(E/Fp, T ) = c(p) +

∞∑n=2

(cpn − pcpn−2)Tn−1.

On the other hand, by definition of the zeta function of E/Fp, one has

−P′

P(E/Fp, T ) =

∞∑n=1

(pn − 1− |E(Fpn)|)Tn−1,

and the last assertion follows. �

9.6. Corollary. One has |E(Fp)| = p+ 1 if p ≡ 3, 5, 6(mod 7), and

|E(Fp)| ≡ 0, 16, 8(mod 28) if p ≡ 1, 2, 4(mod 7) respectively (p 6= 2).

Proof. We already saw the first property. Regarding the second, it follows from thedefinition of ap = Reπp that

cp ≡ 2, 1, 4(mod 7) if p ≡ 1, 2, 4(mod 7).

From this and Proposition 9.3 we deduce

|E(Fp)| ≡ 0, 2, 1(mod 7) if p ≡ 1, 2, 4(mod 7),

Now observe that a model of E is given by the equation

y2 = 4x3 + 21x2 + 28x.

Here, the group E[2] of points of order 2 belongs to E(K), since

4x3 + 21x2 + 28x = 4x(x−√−7 α

4

4)(x+

√−7 ᾱ

4

4),

and E[2] reduces to a subgroup of order 4 of E(Fp). �

10. Jacobians and reduction over finite fields

In this section, if X is a curve defined over k and if k′ is an extension of k, wedenote by X/k′ the curve over k′ obtained from X by extension of scalars.

The quartic Φ4 has discriminant 314.77, hence, the Klein quartic has good reductionfor any prime different from 3 and 7. We denote by X/k the curve defined overa field k with equation Φ4 = 0 (the coefficients being assumed in k!) First, weborrow some information on the number of points of X/k, when k is a finite field.We denote by P (X/k, T ) the numerator of the zeta function of a curve X/k.


10.1. Proposition. If q 6≡ 1(mod 7), then|X(Fq)| = q + 1.

If p ≡ 3 or 5 (mod 7), then P (X/Fp, T ) = 1 + p3T 6, and

|X(Fpm)| = pm + 1 + 6pm/2 if m ≡ 0 (mod 6).If p ≡ 6 (mod 7), then P (X/Fp, T ) = (1 + pT 2)3 and

|X(Fpm)| = pm + 1 + 6pm/2 if m is even.

Proof. Denote by F7 the Fermat curve of degree 7. The map

f(x : y : z) = (x3z : y3x : z3y)

is a dominant morphism from F7 to X, and the map

g(x : y : z) = (x3y : y3z : z3x)

is a dominant morphism from X to the line D with equation x+ y + z = 0. Then

g ◦ f(x : y : z) = (x7 : y7 : z7).If q 6≡ 1(mod 7), then λ 7→ λ7 is an permutation of Fq, hence, g ◦ f induces apermutation of P2(Fq), and this proves the first equality. The map

h(x : y : z) = (x3y : xyz2 : −z3x)induces an isomorphism defined over k from X to the curve

y7 = x(z − x)2z4.Now the two other assertions are consequences of the Stickelberger Relation, as itis proved in [13, Lem. 1.1] or [18, Ex. 12, p. 226]. �

If k is a finite field, the various k-forms of X/k are the curves Y/k defined overk which are isomorphic to X/k over the algebraic closure of k. The k-forms Y/kreflecting at the best the properties of the Klein quartic will be those such thatAutk(Y/k) = G. Since AutC(X/C) is defined over the cyclotomic field L = Q(ζ) oflevel 7, it is worthwile to start from another curve isomorphic over L to the Kleinquartic, namely the quartic C defined over K with equation found by Ciani [5, p.369]

Ψ4(x, y, z) = 0,where

Ψ4(x, y, z) = x4 + y4 + z4 + 3ᾱ(x2y2 + y2z2 + z2x2).This choice is more convenient, since Ciani proved that the group AutC(C/C) isdefined over K. This equation is the one coming from Elkies “S4 model”, and thegenerators of AutC(C/C) are [10, p. 55]

ρ(S) = −

1 0 00 0 10 1 0

and ρ(T ) = 12

−1 1 ᾱα α 0−1 1 −ᾱ

,from which one verifies at once Ciani’s assertion. Recall that an isomorphismX/L→ C/L is provided by

T =

ζ6 + ζ2 αζ + 1 11 ζ6 + ζ2 αζ + 1αζ + 1 1 ζ6 + ζ2

,since Ψ4(T.(x, y, z)) = cΦ4(x, y, z) with a suitable c ∈ L.


Now let k = Fq, where q is a prime power. Remark that there is a couple of distinctsolutions α, ᾱ of the equation x2 + x + 2 = 0 in k if and only if (q/7) = 1, thatis, if q is a square modulo 7. If that is so, one defines successively an isomorphismG

∼−→ Autk(C/k) ⊂ PGL(3, k) by reduction of the isomorphism G∼−→ AutC(C/C)

in characteristic 0, and a plane quartic C/k by reduction of the Ciani quartic C,which has good reduction for any prime different from 7.We shall now show that if q is a square modulo 7, then Jac C/k is isogenous to(E/k)3 (for any curve X, we denote by JacX the Jacobian of X, endowed with itsnatural polarization). One can even be more precise if one uses Hermitian modulesin the style of Serre [36]. Let Mod(Z[α]) be the category of couples (M,H), whereM is an indecomposable Z[α]-module of finite type without torsion, and where H isa positive-definite Hermitian form, such that M is of discriminant 1. We write Min place of (M,H) for simplicity. The group Aut(M) is the group of determinant 1automorphisms of M . Serre proved [23, pp. 235–236] that there is an isomorphismof principally polarized abelian threefolds

Jac X/C ∼ (E/C)⊗Z[α] P,

where P ∈ Mod(Z[α]) is the free module Z[α]3, endowed with the Hermitian form 2 −ᾱ −α−α 2 −1−ᾱ −1 2

.Then Aut(P ) = G, and moreover P is the unique module of rank 3 in Mod(Z[α]),see [10, p. 61]. Turning back to finite fields, one knows [36] that for any curve X/ksuch that JacX/k is isogenous to (E/k)3, there is a unique M ∈ Mod(Z[α]) suchthat

JacX/k ∼ (E/k)⊗Z[α] M.In order to find a curve fulfilling these properties, the curve X/k is not convenient.In fact, according to Torelli’s theorem, see [35] or [23, pp. 236], for any curve Xdefined over k, one has

Autk(JacX/k) = {±1} ×Autk(X/k).Hence, Autk(X/k) must be equal to G and the related representation of G must bedefined over k.

10.2. Theorem. Let k = Fq, where q is a square modulo 7, and C/k the Cianiquartic with equation Ψ4 = 0. Then Autk(C/k) is isomorphic to G, and

Jac C/k ∼ (E/k)⊗Z[α] P.

In particular Jac C/k is isogenous to (E/k)3. Moreover, if q ≡ 1(mod 7), then C/kis isomorphic to the reduction X/k of the Klein quartic.

Proof. First of all, Jac C/k is isogenous to (E/k)3 : an isogeny is explicitly con-structed in [1], or one can use alternately [9, Prop. 2]. This implies, as explainedabove, that

Jac C/k ∼ (E/k)⊗Z[α] M ′

where M ′ ∈ Mod(Z[α]). Now M is of rank 3 and G ⊂ AutM ′, but the only suchmodule is P , as we know. Finally, if q ≡ 1(mod 7), then µ7 ⊂ k and the reductionof the matrix T above is in k. �

Theorem 10.2 has the following consequences. Let k = Fp where p is prime and asquare modulo 7. Since there are fast algorithms for the computation of πp, we havethus obtained a way to calculate fastly the number of points of the Ciani quarticover a finite field:


10.3. Corollary. Let k = Fq, where q is a square modulo 7. Then

P (C/k, T ) = P (E/k, T )3 = |1− πpT |6.Hence,

|C(Fp)| = p+ 1− 3 Tr(πp) = p+ 1− 3cp,and if n ≥ 2, then

|C(Fpn)| = pn + 1− 3 Tr(πnp ) = pn + 1− 3(cpn − papn−2).Moreover if q ≡ 1(mod 7), then |X(Fq))| = |C(Fq)|. �

From Corollary 9.6 we deduce some congruences for that number of points:

10.4. Corollary. One has

|C(Fp)| ≡ 24, 0, 0 (mod 28) if p ≡ 1, 2, 4 (mod 7). �

Considering the prime powers ≤ 1000, with 7 excluded, the maximal number ofpoints is attained in the following cases:

q 4 8 43 107 151 169 683 . . . 213

|X(Fq)| 4 24 80 168 224 248 840 . . . 8736and the minimal number of points in the following cases:

q 2 11 23 32 71 263 331 491 907|X(Fq)| 0 0 0 0 24 168 224 360 728

One can prove, by a method of Serre [10, p. 77], that C/Fq is, for every q in theabove displays, the unique curve of genus 3 over Fq with that number of points.The proof runs as follows. Let X′/Fq be such a curve; since it is maximal (say), ithas the same Frobenius eigenvalues than C/Fq, hence,

Jac X′/Fq ∼ (E/k)⊗Z[α] M ′

with M ′ ∈ Mod(Z[πp]). But one checks that Z[πp] = Z[α] in these cases, hence,M ′ = P since P is the only member of Mod(Z[α]). Then apply Torelli’s Theorem.

Appendix A. Ramanujan’s formulas verbatim

For the convenience of the reader, we reproduce below the last eight lines of [28], p. 300,as edited by Berndt [3]. Recall Ramanujan’s notation:

f(a, b) =

∞Xn=−∞

an(n+1)/2bn(n−1)/2, |ab| < 1,

and

f(−x) = f(−x,−x2) =∞X

n=−∞

(−1)nxn(3n−1)/2, |x| < 1.

The passage begins here (the labels and the equalities inside the brackets have been added):

(A)1

x3/7f(−x3,−x4)f(−x,−x6) − 2 − x

1/7 f(−x2,−x5)f(−x3,−x4) + x

2/7 f(−x,−x6)f(−x2,−x5)

=1

2

(3f(−x1/7)x2/7f(−x7)

+

s4f3(−x1/7)x6/7f3(−x7)

+21f2(−x1/7)x4/7f2(−x7)

+28f(−x1/7)x2/7f(−x7)

)

(B)α2

γ+

β2

α− γ

2

β= 8 +

f4(−x)xf4(−x7)

(C)β

γ2− α

β2− γ

α2= 5 +

f4(−x)xf4(−x7)


[α =v

w, β = −u

v, γ =

w

u]

(D) u = x1/56f(−x3,−x4), v = x9/56f(−x2,−x5), w = x25/56f(−x,−x6)then

(E)u2

v− v

2

w+

w2

u= 0

(F) uvw = x5/8f(−x)f2(−x)

(G)v

u2− w

v2+

u

w2=

f(−x)x7/8f2(−x7)

3

sf4(−x)f4(−x7) + 13x + 49x

2f4(−x7)f4(−x)

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