A Dirichlet Theorem for Rank 1 Elliptic Curves · Elliptic curves carry an addition law which can be described as follows: given two points P;Q, we consider the line through them

A Dirichlet Theorem for Rank 1 Elliptic Curves

Keith Merrill, Brandeis University

June 25, 2020

Keith Merrill, Brandeis University A Dirichlet Theorem for Rank 1 Elliptic Curves

This is based on joint work with Lior Fishman, David Lambert, TueLy, and David Simmons.

My thanks to the organizers and staff for persevering in thesedifficult times and making this conference a success.


Diophantine Approximation

The field of Diophantine approximation can be viewed as anattempt to turn the qualitative aspect of density into aquantitative notion.

As a motivating example, let’s consider the classical situation ofthe density of the rationals in the real numbers. What does itmean to quantify that density?


Dirichlet’s Theorem

Theorem (Dirichlet, Strong Form)

For every α ∈ R and every N ≥ 1, there exists a pair(p, q) ∈ Z× N for which

|qα− p| < 1

N, 1 ≤ q ≤ N.

Corollary (Dirichlet, Weak Form)

For every irrational α, there exist infinitely many rationals p/qsuch that ∣∣∣∣α− p

q

∣∣∣∣ < 1

q2.


We can obviously always find rationals which are closer and closerto α, but doing so generally requires larger and largerdenominators. The idea here is to force rationals with largedenominator to be correspondingly much closer.

The exponent of 2 which appears in the corollary can be interpretedas a measure of the quantitative density of the rationals–indeed,for almost every irrational α, it cannot be replaced by somethingstrictly larger if the corollary is to remain true.


Elliptic Curves

An elliptic curve is formally a smooth curve (a smooth projectivevariety of dimension 1) which has genus 1 and a specifiedbasepoint, which we denote by O.

For our purposes, however, we can consider elliptic curves asdefined by an equation

y2 = x3 + ax + b.

It’s generally more convenient to think of the projectivization ofthe curve given by

{[x : y : z ] : y2z = x3 + axz2 + bz3}, O = [0 : 1 : 0].

We will be concerned with elliptic curves defined over Q, in whichcase we assume a, b ∈ Q.


The set E (C)

We can consider the set of complex points on the curve, thoughtof as sitting inside P2(C).

Given a complex lattice Λ, consider the associated Weierastrass℘-function

℘(z ; Λ) =1

z2+

∑ω∈Λ\{0}

1

(z − ω)2− 1

ω2.

For all z ∈ C \ Λ we have

℘′(z ; Λ)2 = 4℘(z ; Λ)3 − g2℘(z ; Λ)− g3,

where g2 and g3 are quantities associated to Λ.


The set E (C)

Theorem (Silverman, VI, Prop. 3.6b)

Let Λ a complex lattice, and g2 and g3 its associated quantities.Then

E : y2 = 4x3 − g2x − g3.

is an elliptic curve, and the map

ϕ : C/Λ→ E ⊂ P2(C), z 7→ [℘(z ; Λ) : ℘′(z ; Λ) : 1]

is a complex analytic isomorphism of complex Lie groups.

Moreover, by the Uniformization Theorem, every elliptic curvearises in this way. They correspond exactly to complex tori.


The set E (R)

Moreover, if our elliptic curve is defined with real coefficients, thenwe can visualize the set E (R) as the restriction of the map ϕ. Infact, this map is precisely the exponential map of the real locus,viewed as a compact real Lie group.

expE : R→ E (R) ⊂ P2(R), z 7→ [℘(z ; Λ) : ℘′(z ; Λ) : 1]

(Here we need to know that because E is defined over R, Λ isinvariant under complex conjugation, which forces the values℘(z ; Λ) and ℘′(z ; Λ) to be real).

The image of this map is the connected component of the identityE (R)0 and its kernel is of the form Zω for some nonzero period ω.

In fact, E (R) is real-analytically isomorphic to S1 × Φ, where Φhas order 1 or 2.


The set E (R)

Let’s see some pictures. These are courtesy oflmfdb.org[Accessed June 21, 2020]

,


The Group Law

Elliptic curves carry an addition law which can be described asfollows: given two points P,Q, we consider the line through themand take the third point of intersection; we then flip that pointover the x-axis (note that in this presentation the curves areinvariant under reflection). This is defined to be P ⊕ Q.

Image credit to Dylan Pentland of MIT, who credits Silverman.Keith Merrill, Brandeis University A Dirichlet Theorem for Rank 1 Elliptic Curves

The set E (Q)

If our curve has rational coefficients, then it turns out the grouplaw respects points with rational coordinates, in that the sum oftwo points on the curve E with rational coefficients will again be apoint with rational coefficients. While the picture of these points ismaybe not as beautiful as the previous cases, we have the followingincredible theorem:

Theorem (Mordell-Weil)

The group of rational points E (Q) is a finitely-generated abeliangroup.

It follows immediately that we have

E (Q) ∼= Zr ⊕ Ztor ,

where Ztor consists of the torsion points (i.e. finite order). We callr the rank of the elliptic curve.


Rational Approximation on Elliptic Curves

We can now phrase the problem we’re interested in. Given anelliptic curve defined over Q with positive rank, the rational pointson the curve are dense in E (R)0. We want to quantify that density.

We say P ∈ E (R)0 \ E (Q) is κ-approximable if there exists aconstant C and infinitely many Q ∈ E (Q) for which

dist(P,Q) < C (h(Q))−κ.

Here h(·) is the canonical height on E , which defines a positivedefinite quadratic form on the quotient group E (Q)/E (Q)tor , and“dist” is the distance inherited on E from P2(R).


We define the Diophantine exponent of P to be the quantity

νE (P) = lim suph(Q)→∞

− ln(dist(P,Q))

ln(h(Q)),

and we define the Diophantine exponent of the curve E to be

νE = inf{νE (P) : P ∈ E (R)0 \ E (Q)}.

Saying that P is κ-approximable is equivalent to saying thatνE (P) ≥ κ.Waldschmidt has a conjecture (which we will see later) whichwould imply the following:

Conjecture

Let E be an elliptic curve over Q of rank r . Then νE ≥ r2 .


The Main Result

For the rank 1 case, we show stronger than this.

Theorem (Fishman, Lambert, Ly, M., Simmons)

Let E/Q be an elliptic curve with rank 1. Then

νE =1

2.

Even stronger, there exists a constant C such that for everyP ∈ E (R)0 \ E (Q) there exist infinitely many Q ∈ E (Q) satisfying

dist(P,Q) < C (h(Q))−1/2.

This can be interpreted as a uniform weak-type Dirichlet theoremfor the rank 1 case. The second claim of the theorem implies thatνE ≥ 1

2 . For the reverse inequality, we need to show that the set ofbadly approximable points is nonempty.


Sketch Proof

Via the use of the exponential map discussed previously, we canconvert the problem of approximation on E (R)0 to one ofinhomogeneous approximation on the circle.

Let P be an irrational point and let Q be a generator for the freesummand of E (Q). Denote by

θ := exp−1(Q), γ := exp−1(P),

where we choose the unique preimages in the interval (0, ω).

Remark

We have that θω /∈ Q and that there do not exist integers p, q for

which q θω + p = γω .

The first statement follows immediately from the fact that Q hasinfinite order, and the second follows because P is not in the orbitof Q.


Proof–Lower Bound

It turns out that the previous remark is exactly what we need toapply a classical theorem of Minkowski:

Theorem (Minkowski)

Let θω any irrational number and let γ

ω any number for whichγω = p θω + q has no solutions in integers p, q. Then there existinfinitely many pairs of integers (n,m) for which

|n|∣∣∣∣n θω + m − γ

ω

∣∣∣∣ < 1

4.

Since dist(P,Q) differs from minm∈Z |θ + mω − γ| by amultiplicative constant, we have

dist(P, [n]Q) � |nθ + mω − γ| < |ω|4|n|

=C√

h([n]Q).


Proof–Upper Bound

We call a point P badly approximable if there exists a constant c(possibly depending on P) for which

dist(P,Q) > c(h(Q))−1/2

holds for all Q ∈ E (Q). The upper bound νE ≤ 12 will follow if we

can show that there exists a point P ∈ E (R)0 \ E (Q) which isbadly approximable.

Translating as before, it suffices to show that the set{γ = exp−1(P) : ∃c st |nθ + mω − γ| > c

|n|∀n,m

}is non-empty.

Theorem (Bugeaud-Harrap-Kristensen-Velani)

For every θ, ω ∈ R, the set above has full Hausdorff dimension.


Future Directions

This result naturally suggests some follow-up questions.

1 Establish the conjecture for elliptic curves of arbitrary rank.

2 Extend the result to real number fields.

3 Establish Waldschmidt’s conjecture for rank 1 curves. Hisconjecture can be stated as follows:

Conjecture (Waldschmidt)

For every ε > 0 there exists a constant h0 > 0 such that, for anyh ≥ h0 and any P ∈ E (R)0, there exists Q ∈ E (Q) with h(Q) ≤ hand

dist(P,Q) ≤ h−(1/2)+ε.

This is the strong-type of Dirichlet theorem for this setting.


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A Dirichlet Theorem for Rank 1 Elliptic Curves · Elliptic curves carry an addition law which can be described as follows: given two points P;Q, we consider the line through them