[chapter]

Heegner points on elliptic curves of conductor less than 3000

Antoine Gournay

Department of Mathematics and Statistics,

McGill University, Montréal Québec, Canada

June, 2004

A thesis submitted to the Faculty of Graduate Studies and Research

in partial fulfillment of the requirements of the degree of Master of Science

Copyright \begin{matrix}c \\ ◯ \end{matrix} Antoine Gournay, 2004

\underline{i}

Abstract

Elliptic curves are one of the important class of problems in diophantine equations. In this thesis, we describe how to obtain algebraic points (called Heegner points) on elliptic curves. We first go over the necessary definitions and theorems in order to properly study these points and their properties. We pay a particular attention to the property of these point to have integral coordinates or not. We also explain how to make explicit computations of these points and discuss the result of extensive computations made on the elliptic curves of conductor less than 3000, and give a list of the curves that were found to always have Heegner points with integral coordinates.

ii  Abstract

\underline{iii}

Résumé

Les courbes elliptiques sont une classe de problèmes importante d’équation diophantienne. Dans ce mémoire nous décrirons comment il est possible d’obtenir des points algébriques, appelés points de Heegner, sur les courbes elliptiques. Quelques éléments essentiels pour mener à bien la construction de ces points seront d’abord rappelés. On verra comment faire des calculs explicites de ces points, et on discutera ensuite des résultats de ces calculs fait sur les courbes de conducteur inférieur à 3000. Une attention particulière sera réservée l’intégralité des points de Heegner, plus précisement aux courbes dont les points de Heegner ont toujours des coordonnées intégrales.

iv  Résumé

\underline{v}

Acknowledgments

First of all, I would like to thank my supervisor Henri Darmon, for his help, support and enthusiasm these last years. I would also like to thank NSERC for allowing me to pursue my studies in mathematics. I also want to thank McGill University, and in particular the Mathematics and Statistics department and its professors: I learned valuable knowledge from them in all branches of mathematics. I would also like to thank my fellow students and friends, particularly Hugo, for the all the advices and distractions.

vi  Acknowledgments

\underline{vii}

Table of Contents

Abstract i

Résumé iii

Acknowledgments v

Introduction 1

1 Elliptic Curves 3 1.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . 3 1.2 The Group Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Reduction modulo 𝔭 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Complex Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.1 Endomorphisms rings . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.2 Finiteness of Ell(O) . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.3 The action of Pic(O) . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.4 Explicit Galois Action on j(E) . . . . . . . . . . . . . . . . . 22

2 Heegner Points 27 2.1 Modular Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Galois action on Heegner points . . . . . . . . . . . . . . . . . . . . . 34 2.4 Calculation of Heegner Points . . . . . . . . . . . . . . . . . . . . . . 38 2.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.1 Modular uniformisation . . . . . . . . . . . . . . . . . . . . . 41 2.5.2 Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5.3 Trivial points . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.4 The point P_{K} . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Conclusion 45

A Binary Quadratic Forms 47

B List of integral curves 55

\underline{1}

Introduction

The diophantine problem consists in finding solutions over the integers or the rational numbers to polynomial equations. After the linear case (for example ax + by + c = 0 solved by the euclidian algorithm) and the quadratic case (for example Pell’s equation), the next degree of complexity comes with elliptic curves. Like the two first examples where solutions can be seen as an abelian group (albeit a infinite cyclic group), solutions to the diophantine problem of an elliptic curve form an abelian group, via the chord and tangent method. Though it has been shown that elliptic curves over a finite field or a number field K are finitely generated, there is still no reliable method to find the generators.

In the last decades, there emerged the idea that elliptic curves could be parametrized by another object. Looking at certain subgroups of SL_{2}(ℤ) acting by M obius transformations on the complex upper half-plane H, it is possible to form new Riemann surfaces by the usual quotient operation. These surfaces are not compact, however a proper compactification turns them into complete complex algebraic varieties, called modular curves. Thanks to the work of Wiles and all the improvements that have been made to it, we are now aware that a model over ℚ of these varieties admits an algebraic map to a certain class of elliptic curves (see [25], [24], [6], [3], [2] and [7]).

Gross and Zagier used this map to construct points on elliptic curves. The algebraicity of this parametrization gives Galois control on the image of this map. Heegner points are points on the modular curve of which we know that the image by the modular parametrization are going to be algebraic. Apart from being an explicit method to get an algebraic point on a curve, it is also possible to make actual calculations of this point. Though these points have been used to prove important results related to the Birch and Swinnerton-Dyer conjecture, few of them have actually been computed.

In the following pages we will make a brief recall of the concepts and results necessary in order to define properly Heegner points and study their basic properties. In hope of getting more insight on the modular parametrization, explicit computations of such points have been made on all the elliptic curves of conductor less than 3000. A question that is of particular concern to us, is whether coordinates of the Heegner points obtained are algebraic integer or not, and if this is not the case, does the modular parametrization maps certain points of the modular curve (non-cuspidal point) to the point at infinity of the elliptic curve.

Chapter 1

Elliptic Curves

This chapter begins by a very brief recall of the concepts that are necessary to define Heegner points. A complete discussion of these subjects can be found in [15] and [22]. However, we will go more leisurely over the theory of complex multiplication for it is crucial in the constructions we wish to make.

1.1 Definitions and Basic Properties

Definition 1.1. An elliptic curve E over a field F (denoted E_{/F}) is an algebraic curve of genus 1, possessing a specified base point.

One can show ([22] III.§3) that an elliptic curve over F is isomorphic to the zero locus in ℙ^{2}(\overline{F}) of an equation

E : Y^{2}Z + a_{1}XY Z + a_{3}Y Z^{2} = X^{3} + a_{2}X^{2}Z + a_{4}XZ^{2} + a_{6}Z^{3}  (1.1)

where a_{i} ∈ F and the specified base point is the “point at infinity” denoted O_{E} = [0, 1, 0]. This is known as the Weierstrass equation of E. When K ⊇ F, we write E(K) to denote the zero locus of the same equation in ℙ^{2}(K). To ease notation, it is convenient to write x = X/Z and y = Y/Z, O_{E} being the only point which cannot be described by (x, y). If char(F) ≠ 2 then one can replace y by \frac{1}{2}(y − a_{1}x − a_{3}) to get

E : y^{2} = x^{3} + b_{2}x^{2} + b_{4}x + b_{6}
where b_{2} = a_{1}^{2} + 4a_{2}, b_{4} = 2a_{4} + a_{1}a_{3}, and b_{6} = a_{3}^{2} + 4a_{6}

(1.2)

Definition 1.2. Let b_{8} = a_{1}^{2}a_{6} + 4a_{2}a_{6} − a_{1}a_{3}a_{4} + a_{2}a_{3}^{2} − a_{4}^{2}, then we define

\begin{aligned}& Δ = −b_{2}^{2}b_{8} − 8b_{4}^{3} − 27b_{6}^{2} + 9b_{2}b_{4}b_{6} \\ and & ω = dx/(2y + a_{1}x + a_{3}) = dy/(3x^{2} + 2a_{2}x + a_{4} − a_{1}y) \end{aligned}  (1.3)

as respectively the discriminant and the invariant differential (unique up to scalar multiplication) of the Weierstrass equation, and

j = (b_{2}^{2} − 24b_{4})^{3}/Δ  (1.4)

as the j-invariant of the elliptic curve E.

One can check that the only change of variables that preserve the Weierstrass form and fix O_{E} = [0, 1, 0] are of the form x = u_{0}^{2}x′ + u_{2} and y = u_{0}^{3}y′ + u_{0}^{2}u_{1}x′ + u_{3} for u_{0}, u_{1}, u_{2}, u_{3} ∈ K and u_{0} ≠ 0. When two equations are related by such a transformation we say that they are equivalent, or that the 2 curves they describe are isomorphic over K ⊇ F . The quantities we just defined change as follows:

Δ = u_{0}^{12}Δ′, u_{0}ω = ω′ and j = j′  (1.5)

In particular, if u_{i} = 0 for i = 1, 2, 3 then one has that u_{0}^{j}a′_{j} = a_{j} and u_{0}^{j}b′_{j} = b_{j}. This makes the j-invariant stand out as a quantity independent of the equation chosen to represent E. In fact, if two curves have the same j-invariant then they are isomorphic over \overline{F} as algebraic curves. Also, since E has genus one, ω is the only holomorphic non-vanishing differential, up to scalar multiplication, attached to the Weierstrass equation. Simple calculations ([22] III.§1) reveal that Δ = 0 if and only if the curve E is singular.

Actually if P = (x_{0}, y_{0}) is a singular point, it is instructive to consider the Taylor expansion of the curve equation at that point. Let f(x, y) = y^{2} + a_{1}xy + a_{3}y − x^{3} − a_{2}x^{2} − a_{4}x − a_{6}, then f(x_{0}, y_{0}) = 0, \frac{∂f}{∂x}(x_{0}, y_{0}) = 0 and \frac{∂f}{∂y} (x_{0}, y_{0}) = 0. Thus,

f(x, y) − f(x_{0}, y_{0}) = ((y − y_{0}) − α(x − x_{0}))((y − y_{0}) − β(x − x_{0})) − (x − x_{0})^{3} (1.6)

where α and β are at most in a quadratic extension of F. We distinguish two types of singular points:

Definition 1.3. We say P is a node (resp. cusp) if α ≠ β (resp. α = β), it then has two tangent lines y − y_{0} = α(x − x_{0}) and y − y_{0} = β(x − x_{0}) (resp. one tangent line y − y_{0} = α(x − x_{0})).

Further inspection of these cases show that E has a unique cusp when Δ = b_{2}^{2} − 24b_{4} = 0 and a unique node when Δ = 0, b_{2}^{2} − 24b_{4} ≠ 0.

Lastly, since elliptic curves are algebraic varieties with a specified point it makes sense to define a more refined class of morphisms:

Definition 1.4. Let E_{/F} and E′_{/F} be two elliptic curves. An isogeny of E to E′ over K ⊇ F is a non-constant morphism f : E → E′ defined over K such that f(O_{E}) = O_{E′}. Two curves E and E′ are said to be isogenous if such an isogeny exists over \overline{F}.

Since morphisms of curves are either constant or surjective (with each point having a finite preimage), either f(E) = {O_{E′}} or f(E(K)) = E′(K). Like all morphism of curves we can define the (separable, inseparable) degree of an isogeny as the (separable, inseparable) degree of the extension K(E)/f^{∗}K(E′) where K(E) is the function field of E. Over fields of characteristic p > 0, if the pullback of a differential by an isogeny is 0, then it is inseparable. Also, if the curve is non-singular and the isogeny is inseparable, then f = λ∘F r where F r is the q^{th}-power Frobenius map (q = deg_{i}(f) is a power of p), and λ is separable (see [22] II and [12]).

1.2 The Group Law

One of the crucial properties of elliptic curves is that one can explicitly construct a point on the curve from two other ones. On the real numbers, the method is simple. First make a change of variables to get E : y^{2} = x^{3} + b_{2}x^{2} + b_{4}x + b_{6}. Then the line through two points (say P and Q) on the curve intersects a third point (if one consider the point at infinity to be on all the lines with x constant). The reflection of this point by the x-axis is called P + Q. We can extend this construction when P = Q by taking the tangent at P. Interestingly, we can express the coordinates P + Q explicitly as an algebraic formula on the coordinates of P and Q (see [22] III.2). As a consequence, we can extend this construction to any number field F, and the resulting point is always defined on F. Note that these formulas can be found without making a change of variables so that even when char(F) = 2, the points can be combined.

The most interesting property of the construction + is that it defines an abelian group law on the points E(F) with identity element O. This law also enables us turn the set of isogenies from A to A′ over K, into a group Hom_{K}(A, A′) with the law (f + g)(P) = f(P) + g(P) for f, g isogenies. As usual, we define the ring of endomorphism End_{K}(A) = Hom_{K}(A, A) by setting multiplication as composition.

It is important to point out that the group law on E gives a map from ℤ to End_{K}(E). For n ∈ ℤ_{≥0} we define [n](P) = P + ⋯ + P ( n times ) , and for n ∈ ℤ_{<0}, [n]P = [−n](−P). This is always an injection, and in fact End_{K}(E) is ℤmodule of rank one, two or four. Over fields of characteristic 0, elliptic curves with endomorphism ring bigger than ℤ have many additional properties, and the last section of this chapter is devoted to their study.

Before going on, we will state 3 important theorems that explain the structure of the group of E:

Theorem 1.1. (Singular curves) Let E_{/F} be a elliptic curve defined over a field F by a singular Weierstrass equation. If the singular point on E is a cusp, then E(F) ≃ F^{+}. If the singular point is a node, let α, β be the slopes of the tangent lines at the node, and set K = F (α) = F (β). If K = F then E(F) ≃ F^{×}. If [K : F] = 2 then E(F) ≃ {x ∈ K^{×} | N_{K/F} (x) = 1}.

Theorem 1.2. (Mordell-Weil) Let E be a non-singular elliptic curve defined over a number field F. The abelian group E(F) is finitely-generated. It’s rank over ℤ is called the rank of E over F.

Theorem 1.3. (Weierstrass uniformisation) Let E be a non-singular elliptic curve defined over ℂ, then there is a lattice Λ ⊆ ℂ (i.e. a discrete subgroup of (ℂ, +) of rank 2 over ℤ) such that there is a complex analytic Lie group isomorphism of E(ℂ) with ℂ/Λ. The isomorphism is defined as follows: let ω_{E} be the invariant differential of E, let Ω_{1} and Ω_{2} be two closed paths on E(ℂ) that generate the fundamental group, let ω_{i} = ∫_{Ω_{i}} ω_{E}, and let Λ_{E} = ℤω_{1}⊕ℤω_{1} be the Néron lattice of E. Then the isomorphism is given by

\begin{matrix}Φ_{w}^{−1} : E(ℂ) & → ℂ/Λ \\ P & ↦ ∫_{O}^{P} ω_{E} (mod Λ) \end{matrix}  (1.7)

Conversely, if Λ is lattice in ℂ, then

\begin{matrix}Φ_{w} : ℂ/Λ & → E(ℂ) : y^{2} = 4x^{3} − g_{2}(Λ)x − g_{3}(Λ) \\ z & ↦ Φ_{w}(z) = (℘_{Λ}(z), ℘′_{Λ}(z)) \\ 0 & ↦ O \end{matrix}  (1.8)

where ℘_{Λ}(z) = \frac{1}{z^{2}} + ∑_{λ∈Λ−\{0\}} (\frac{1}{(z−λ)^{2}} − \frac{1}{λ^{2}} )
and g_{2}(Λ) = 60 ∑_{λ∈Λ−\{0\}} \frac{1}{λ^{4}}

g_{3}(Λ) = 140 ∑_{λ∈Λ−\{0\}} \frac{1}{λ^{6}}

(1.9)

is the inverse isomorphism, with Φ_{w}^{∗}(dx/y) = dz.

The full strength of the last theorem is summarized as the equivalence of the following categories:

\{ \begin{matrix}Elliptic curves over ℂ \\ with isogenies \end{matrix} \} ≃ \{ \begin{matrix}tori ℂ/Λ with analytic \\ maps sending 0 to 0 \end{matrix} \}  (1.10)

This equivalence will be very useful to study End_{K}(E), as isogenies are more difficult to characterize a priori than analytic maps from ℂ/Λ to itself.

Since all curves isomorphic over ℂ have the same j-invariant, it is convenient to denote the j-invariant of E by j(τ) where the lattice Λ associated to E is homothetic to ⟨1, τ⟩ := ℤ ⊕ ℤτ, and ℑ(τ) > 0. Similarly Δ(τ) stands for the discriminant of the equation of the curve Φ_{w}(ℂ/⟨1, τ⟩). Note that the choice of τ is made up to the action of SL_{2}(ℤ) as described by A.4.

1.3 Reduction modulo 𝔭

Weierstrass uniformisation shows us that we gain insight on curves defined over a number field by looking at them over a larger field (namely ℂ). One could then be tempted to do the same but looking at “smaller” fields, namely the finite fields obtained by looking at the algebraic integers modulo some prime ideal. A naive idea would be to look at the Weierstrass equation modulo some prime 𝔭. Of course, for such a thing to be possible we first have to make sure the a_{i} are integers in F. Since the change of variables ϕ : (x, y) ↦ (u^{−2}x, u^{−3}y) multiplies each a_{i} by u^{i}, it is always possible to do so. However, taking u ∈ 𝔭^{k} for k big enough would make all the coefficients reduce to 0 (mod 𝔭), so we have to find a proper equation before reducing.

Definition 1.5. Let E_{/F} an elliptic curve defined over F, R be the ring of integers of F, 𝔭 be a prime ideal of R, and ν_{𝔭} be the valuation at 𝔭. The Weierstrass equation

1.3 Reduction modulo 𝔭  9

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6} is said to be minimal for a prime 𝔭 of R if of all equations y^{2} + a′_{1}xy + a′_{3}y = x^{3} + a′_{2}x^{2} + a′_{4}x + a′_{6} describing the curve E with a′_{i} ∈ R for i = 1, 2, 3, 4, 6, ν_{𝔭}(Δ(a_{i})) ≤ ν_{𝔭}(Δ(a′_{i})).

Note that the possible values of ν_{𝔭}(Δ) form a discrete set (a_{i} ∈ R), thus it takes a minimal value. Once we have found this minimal equation, we can define the reduction of E at 𝔭 (written Ẽ_{𝔭}) as the curve over R/𝔭 whose equation has the same coefficients but modulo 𝔭. Amongst many thing, this reduced elliptic curve helps to determine the torsion on the original curve (see [22] V). Sometimes, it may happen that for a given prime the valuation of Δ_{E} is greater than 0 even when E is described by the minimal equation. In this case, the reduced curve is singular, Δ ≡ 0 (mod 𝔭). For these primes we say that E has bad reduction.

Definition 1.6. Let E an elliptic curve defined over F, and 𝔭 a prime for which the reduction of E at 𝔭 is a singular curve. We say that E_{/F} has bad reduction at 𝔭. Furthermore,

1. if Ẽ_{𝔭} has a cusp then E_{/F} has additive bad reduction at 𝔭.

2. if Ẽ_{𝔭} has a node and Ẽ_{𝔭}(R/𝔭) ≃ (R/𝔭)^{×} then E_{/F} has split multiplicative reduction at 𝔭.

3. if Ẽ_{𝔭} has a node and Ẽ_{𝔭}(R/𝔭) ≃ / (R/𝔭)^{×} then E_{/F} has non-split multiplicative reduction at 𝔭.

Finally, when Ẽ_{𝔭} is non-singular we say that E_{/F} has good reduction at 𝔭.

Note that since Δ is only divisible by finitely many primes, the number of primes with bad reduction is always finite for a given elliptic curve. There is a ideal that one usually associates to an elliptic curve E_{/F} to encode the bad and good reduction. Though a proper definition would require the introduction of many concept from the representation theory of local fields (see [23] IV.§10), we can almost always reduce to the following one:

Definition 1.7. The arithmetic conductor of a curve E_{/F} is 𝔑 = ∏_{prime 𝔭◁R}𝔭^{f(𝔭)}, where f(𝔭)

is 0 when E_{/F} has good reduction at 𝔭

is 1 when E_{/F} has multiplicative reduction at 𝔭 is 2 when E_{/F} has additive reduction at 𝔭 and 𝔭 | p ≥ 5 is greater or equal to 2 otherwise.

When K = ℚ we define the conductor to be the positive integer such that (N) = 𝔑.

Note that both definition 1.6 and 1.7 depend heavily on the field over which the curve is defined.

One of the interesting properties of reducing elliptic curves is that when the reduction is good, the degree of the isogenies remain unchanged. Another interesting quality of the reduced curve Ẽ_{𝔭}(R/𝔭) is its number of elements, n_{𝔭}. As we do not expect a cubic polynomial to favor squares over non-squares in its value, it seems reasonable to give |𝔭| + 1 := N_{F/ℚ}(𝔭) + 1 as an estimate of n_{𝔭} (when the reduction is good), as every square value of the polynomial give two root, and we expect them to happen 50% of the time (|𝔭| = ♯R/𝔭). We then define a_{𝔭} = |𝔭| + 1 − n_{𝔭} as our error term. However, when E_{/F} has bad reduction at 𝔭, our estimate should be |𝔭| as we have to take out the singular point, and one can find, thanks to theorem 1.5, that a_{𝔭} is 0 (resp. 1, -1) when E_{/F} has additive (resp. split, non-split multiplicative) reduction.

We can now define the L-series of E_{/F} as

L(E_{/F} , s) = ∏_{prime 𝔭◁R}L_{𝔭}(E_{/F} , |𝔭|^{−s})^{−1}  (1.11)

where L_{𝔭}(E_{/F} , X) = 1 − a_{𝔭}X + |𝔭|X^{2} when 𝔭 has good reduction
L_{𝔭}(E_{/F} , X) = 1 − a_{𝔭}X  when 𝔭 has bad reduction

(1.12)

Note that L_{𝔭}(E_{/F} , |𝔭|^{−1}) = n_{𝔭}/|𝔭|.

1.4 Complex Multiplication

As we observed in the first section of this chapter, the nature of the group structure that we defined on an elliptic curve enables us to embed ℤ in the ring of endomorphisms of the curve. Though in most cases this map is also surjective, there also exist curves with endomorphism ring bigger than ℤ. We will devote the rest of this chapter to the study of these curves, which will be henceforth referred to as CM curves or curves with complex multiplication.

1.4.1 Endomorphisms rings

Our first step will be to attempt to describe End(E). As we are only concerned about fields of characteristic 0, it will be enough to do it over ℂ. The general case will follow from the Lefschetz principle (see [22] VI.§6). We can reduce to an easier problem using Weierstrass uniformisation. Indeed, since elliptic curves over ℂ are isomorphic to ℂ/Λ (as Riemann surfaces), where Λ = ⟨ω_{1}, ω_{2}⟩ is a lattice, it is sufficient to look at their endomorphism ring.

First, let’s show that End(ℂ/Λ) = {α ∈ ℂ | αΛ ⊆ Λ}. If α ∈ ℂ is such that αΛ ⊆ Λ then f(z) = αz ∈ End(ℂ/Λ).

Conversely, take f ∈ End(ℂ/Λ), then f is an analytic map sending 0 to 0. There is a projection p (so also a covering map) from ℂ to ℂ/Λ. From algebraic topology, namely the general lifting lemma, {1} = (f ∘ p)_{∗}(π_{1}(ℂ)) ⊆ p_{∗}(π_{1}(ℂ)) = {1}, where π_{1} is the fundamental group, guarantees the existence of a lift of f : ℂ/Λ → ℂ/Λ to f̃: ℂ → ℂ, such that the diagram commutes:

\begin{matrix}ℂ & ⟶^{f̃} & ℂ \\ ^{p}↓ & & ^{p}↓ \\ ℂ/Λ & ⟶^{f} & ℂ/Λ \end{matrix}  (1.13)

But f̃ is easy to describe:

p ∘ f̃(z + λ) = p ∘ f̃(z)  ∀z ∈ ℂ, ∀λ ∈ Λ
⇒  f̃(z + λ) ≡ f̃(z) (mod Λ) ∀z ∈ ℂ, ∀λ ∈ Λ
⇒  f̃(z + λ) −f̃(z) ∈ Λ  ∀z ∈ ℂ, ∀λ ∈ Λ

(1.14)

However Λ is a discrete set, and f̃(z + λ) − f̃(z) is a holomorphic function, so it must be a constant possibly depending on λ. Taking the derivative, one has that f̃′(z+λ)−f̃′(z) = 0 ∀λ ∈ Λ which means that f̃′(z) is bounded on ℂ, and consequently also a constant (by Liouville’s theorem). We can then conclude that f̃(z) = αz + β. Finally since f̃(0) ∈ Λ, we get that f(z) = αz + β with αΛ ⊆ Λ and β ∈ Λ. Without loss of generality we can assume that β = 0

So End(E) = O_{Λ} = {α ∈ ℂ | αΛ ⊆ Λ}. Taking τ = ω_{2}/ω_{1} of positive imaginary part, and replacing Λ by ⟨1, τ⟩ (as they are homothetic) we find that for any α ∈ O_{Λ} there are a, b, c, d ∈ ℤ with α = a + bτ and ατ = c + dτ. Simple manipulations give that α is integral over ℤ as α^{2} − (a + d)α + bc = 0. Now since O_{Λ} ≠ ℤ, we have that for an α ∈| ℤ the corresponding b ≠ 0 so bτ^{2} + (d − a)τ + c = 0. Thus ℚ(τ) is a quadratic extension, and as τ ∉ ℝ (else Λ is not lattice) it is also imaginary. Finally since O_{Λ} ⊆ ℚ(τ) is a ring and is integral over ℤ, O is an order in a quadratic field. Theorem 1.4. Let K be a field of characteristic 0, and E an elliptic curve over K with endomorphism ring different from ℤ (a CM curve). Then End(E) is an order in a quadratic field.

There is actually a preferred identification of End_{ℂ}(E) with O_{Λ}. The lattice identification of O_{Λ} with End_{ℂ}(E) consists of associating α ∈ O_{Λ} with the endomorphism [α] such that for any invariant differential ω of E, [α]^{∗}ω = αω.

The idea to fix such an identification is simple. To α ∈ O one associates the endomorphism [α] such that the following diagram commutes:

\begin{matrix}ℂ/Λ & ⟶^{f_{α}} & ℂ/Λ \\ ^{Φ_{w}}↓ & & ^{Φ_{w}}↓ \\ E & ⟶^{[α]} & E, \end{matrix}  (1.15)

where f_{α} is defined by z ↦ αz. To prove that [α]^{∗}ω = αω, we first note that both any two invariant differentials on ℂ/Λ are equal up to multiplication by a non zero constant, as their quotient is a function invariant under translation on ℂ. Now tracing through the diagram, [α] = f ∘ f_{α} ∘ f^{−1} so

\begin{aligned}[α]^{∗}(ω) & = (Φ_{w}^{−1})^{∗} ∘ f_{α}^{∗} ∘ Φ_{w}^{∗}(ω) \\ & = (Φ_{w}^{−1})^{∗} ∘ f_{α}^{∗}(c dz) = (Φ_{w}^{−1})^{∗}(αc dz) \\ & = αω \end{aligned}  (1.16)

Example 1.1. Here are two classical examples of elliptic curves (over ℂ) with complex multiplication:

1. E : y^{2} = x^{3} + x with End(E) = ℤ[i] and [i](x, y) = (−x, iy)

2. E : y^{2} = x^{3} + 1 with End(E) = ℤ[ρ] and [ρ](x, y) = (ρx, y), where ρ = e^{2πi/3}

1.4.2 Finiteness of Ell(O)

Rather than concentrate on a particular elliptic curve with CM by O, it is useful to look at all the elliptic curves with CM by some given order. We will further denote by K = O ⊗_{ℤ} ℚ the fraction field of O lies, and by O_{K} the maximal order of K.

For 𝔞 a lattice in ℂ let us denote O_{𝔞} = End(ℂ/𝔞) = {α ∈ ℂ | α𝔞 ⊆ 𝔞}. Furthermore, for O ⊆ K define

Ell(O) := {elliptic curves with CM by O}/isomorphisms over ℂ
= {j(E) | E has CM by O}
= {lattices 𝔞 ⊆ ℂ | O_{𝔞} = O}/ℂ^{×}
= {lattices 𝔞 ⊆ K | O_{𝔞} = O}/K^{×}

(1.17)

where the meaning of = is that there is a one to one correspondance. The third equality comes from the fact that two complex tori are isomorphic is and only if their lattices Λ, Λ′ are homothetic (i.e. ∃α ∈ ℂ such that Λ = αΛ′.The last equality holds since if 𝔞 = ⟨1, τ⟩ and O_{𝔞} = O = ℤ + cℤω (where c is the conductor of O) then cωτ = a + bτ for some a, b ∈ ℤ and thus τ ∈ ℚ(ω) = K.

It is consequently natural to pursue our studies by looking at the set Lat(O) = {lattices 𝔞 ⊆ K | O_{𝔞} = O}.

Lemma 1.1. If 𝔞 is an invertible fractional O-ideal, then 𝔞 is in Lat(O).

Proof. Clearly, O ⊆ O_{𝔞}. On the other hand if α ∈ ℂ is such that α𝔞 ⊆ 𝔞 then αO = α𝔞𝔞^{−1} ⊆ 𝔞𝔞^{−1} = O. So α ∈ O.

As a consequence of this lemma, we get that the ideal class group of O, Cl(O) = {invertible fractional O-ideals}/K^{×} injects in Lat(O)/K^{×} = Ell(O). As usual one is led to wonder how much of the target is missed by such a map, which is the goal of our next lemma:

Lemma 1.2. Given a lattice 𝔞 ⊆ K such that O_{𝔞} = O then 𝔞 is an invertible fractional O-ideal

Proof. Without loss of generality, let 𝔞 = ℤ + ℤτ for τ in K. τ has a minimal polynomial of the form Ax^{2} + Bx + C with gcd(A, B, C) = 1. Let D = B^{2} − 4AC, and τ̄ be the Galois conjugate of τ. We have τ + τ̄ = −B/A and τ · τ̄ = C/A, so if we define 𝔞̄ =< 1, τ̄ >, we get

\begin{aligned}𝔞𝔞̄ & = ⟨1, τ, τ̄, ττ̄⟩ \\ & = ⟨1, τ, τ̄ + τ, C/A⟩ \\ & = ⟨1, (−B + \sqrt{D})/2A, −B/A, C/A⟩ \\ & = A^{−1}⟨A, (−B + \sqrt{D})/2, B, C⟩ \\ & = A^{−1}⟨1, (B + \sqrt{D})/2⟩ \end{aligned}  (1.18)

The last equality holding because gcd(A, B, C) = 1. Now if we have that

O = ⟨1, (B + \sqrt{D})/2⟩  (1.19)

then our proof is over as A𝔞̄ is an inverse to 𝔞.

Let’s denote ω = (B + \sqrt{D})/2 and M = ⟨1, ω⟩. Since 1τ ∈ 𝔞 and ωτ = −(B^{2} − D)/4A = C ∈ ℤ ⊆ 𝔞, M ⊆ O_{𝔞} = O. Now for a z ∈ O = O_{𝔞},

z𝔞 ⊆ 𝔞 ⇒ z𝔞(\overline{𝔞}A) ⊆ 𝔞(\overline{𝔞}A) ⇒ zM ⊆ M  (1.20)

In particular z belongs to M. Therefore M = O.

Putting the two lemmas together we get our first significant result

Theorem 1.5. Ell(O) ≃ Cl(O)

So in particular, since Cl(O) is finite (see lemma A.1 or [14]) we have that Ell(O) is also finite. An important observation for the proper use of this result is that given an elliptic curves E : y^{2} = x^{3} + ax + b over ℂ with CM by an order O, then for any automorphism σ of ℂ, σE : y^{2} = x^{3} + (^{σ}a)x + (^{σ}b) possess the same endomorphism ring, by the identification f ↦ (σf), where σf is obtained from f by applying σ to the coefficients of the morphism.

Lemma 1.3. Let A be an elliptic curve with CM by O, and let [·]_{A}, [·]_{^{σ}A} denote the respective lattice identifications of O. Then^{σ}([α]_{A}) = [^{σ}(α)]_{^{σ}A} ∀α ∈ O and ∀σ ∈ Aut(ℂ). Furthermore, if A is defined over L and K = Quot(O) then every endomorphism of E is defined over L · K

Proof. Let ω_{E} denote the invariant differential of the elliptic curve E. Then one notes that ^{σ}ω_{A} = ω_{^{σ}A} (by Def 1.2) and thus

\begin{aligned}(^{σ}[α]_{A})(ω_{^{σ}A}) & =^{σ} ([α]_{A}^{∗}ω_{A}) & =^{σ} (αω_{A}) \\ & =^{σ} αω_{^{σ}A} & = [^{σ}α]_{^{σ}A}ω_{^{σ}A} \end{aligned}

Using the fact that the pullback is an injection of End(E) into the endomorphism of the space of invariant differentials of E (see [22] II.§4), we conclude that ^{σ}([α]_{A}) = [^{σ}(α)]_{^{σ}A}

If A possesses an equation over L, and σ ∈ Aut(ℂ) fixes L, then ^{σ}A = A. So in particular, we get that ^{σ}([α]_{A}) = [^{σ}α]_{A}. If σ also fixes K, then ^{σ}([α]_{A}) = [α]_{A} and so [α]_{A} can be defined over L · K

The fact that the endomorphism ring remains unchanged when we apply an automorphism of ℂ to an elliptic curve, gives us a way to produce other curves with CM from an existing one. Using this idea one can already find useful results:

Theorem 1.6. Let E over ℂ be a CM curve, then j(E) is an algebraic number.

Proof. If E is a CM curve, then E ∈ Ell(O) for some O. The algebraic definition of the j-invariant (see definition 1.2) makes it clear that ^{σ}j(E) = j(^{σ}E), thus if j(E) is transcendental then there are infinitely many automorphisms of ℂ not fixing j(E), giving infinitely many non-isomorphic (since their j-invariant is different) elliptic curves with CM by O, contradicting the finiteness of Ell(O).

Since j(E) is algebraic, it seems reasonable to ask what is the field generated by all the j-invariants of the curves in O. This theorem also exhibits an action of the Galois group G_{\overline{K}/K} on the set Ell(O) via the j-invariants, which deserves a deeper inspection.

But first, let us define another action of Cl(O), (which is related to G_{K/ℚ} by class field theory) in the hope of getting more insights.

1.4.3 The action of Pic(O)

To do so, we need a classical result, namely that Cl(O) is isomorphic to

Pic(O) = {rank one projective O-module, up to isomorphisms}.  (1.21)

The group law on the latter is defined as follows:

(𝔞, 𝔟) ↦ 𝔞 ⊗_{O} 𝔟  (1.22)

The identity is O and the inverse of 𝔞 is Hom_{O}(𝔞, O). The isomorphism Cl(O) → Pic(O) is then described as [𝔞] ↦ [𝔞], and is a group homomorphism since 𝔞 ⊗_{O} 𝔟 = 𝔞𝔟 ⊗_{O} O. The proof of this can be found in [13]. This isomorphism is very useful to define an action of Pic(O) on the set Ell(O).

Let A ∈ Ell(O), and 𝔞 ∈ Cl(O) such that 𝔞 is coprime to the conductor of O, so that 𝔞 is invertible in O. Since it is always possible to choose such an ideal in every equivalence class (see remark A.1), we can define the action of Pic(O) on Ell(O) as

𝔞 ⋆ A = Hom_{O}(𝔞, A)  (1.23)

for 𝔞 an invertible ideal. However this only makes sense if Hom_{O}(𝔞, A) is an elliptic curve. To see this, we will consider A(ℂ) = ℂ/Λ. Then we look at the following exact sequences:

0 →  Λ  →  ℂ  →  ℂ/Λ  → 0
0 → Hom_{O}(𝔞, Λ) → Hom_{O}(𝔞, ℂ) → Hom_{O}(𝔞, ℂ/Λ)

(1.24)

But the second one is also right exact since 𝔞 is a projective O-module (for it is invertible). To make explicit the structure of modules of the form Hom_{O}(𝔞, −) we need some lemmas:

Lemma 1.4. Let 𝔞 be an invertible O-ideal. Then the ideal 𝔞_{𝔭} = 𝔞O_{𝔭} is principal for every prime 𝔭 of O.

Proof. Let 𝔟 be such that 𝔞𝔟 = O, then for some a_{i} ∈ 𝔞 and b_{i} ∈ 𝔟, where i = 1, … k, k ; ∑_{i=1}a_{i}b_{i} = 1. Thus it is impossible for all a_{i}b_{i} to be in 𝔭O_{𝔭}. Suppose a_{1}b_{1} ∉ 𝔭O_{𝔭}. Then it is a unit. Take any a ∈ 𝔞, then b_{1}a ∈ 𝔟𝔞_{𝔭} = O_{𝔭}. It follows that ab_{1}(b_{1}a_{1})^{−1}a_{1} ∈ a_{1}O_{𝔭} so 𝔞_{𝔭} = a_{1}O_{𝔭}.

Lemma 1.5. Let 𝔞 be an invertible O-ideal, and M a torsion-free O-module. Then the map

\begin{aligned}f : & 𝔞^{−1} ⊗_{O} M & → & Hom_{O}(𝔞, M) \\ & α ⊗_{O} m & ↦ & (ϕ_{α,m} : x ↦ (αx)m), \end{aligned}  (1.25)

extended by O-linearity is an isomorphism

Proof. It is trivial to show that this is a group homomorphism. Bijectivity requires more work; we need to study the localization at every prime of O. To do so, recall that there is a natural isomorphism (O_{𝔭} is the localization of O at 𝔭)

\begin{matrix}O_{𝔭} ⊗_{O} Hom_{O}(𝔞, M) & ≃ & Hom_{O_{𝔭}}(O_{𝔭} ⊗_{O} 𝔞, O_{𝔭} ⊗_{O} M) \\ (β, f) & ↦ & (ϕ : a ⊗_{O} m ↦ βa ⊗_{O} f(m)) \end{matrix}  (1.26)

under the hypothesis that 𝔞 is finitely presented. But the sequence

0 → 𝔞^{−1}(α_{2}, −α_{1}) → O ⊕ O → 𝔞 → 0  (1.27)

where 𝔞^{−1}(α_{2}, −α_{1}) = {(aα_{2}, −aα_{1}) for a ∈ 𝔞^{−1}}, is exact. But since 𝔞^{−1} is also a fractional ideal, it has at most two generators, thus is contained in O ⊕ O, so 𝔞 is finitely presented. Alternatively, we recall that all finitely generated modules over noetherian rings are finitely presented.

Using 1.25 we can localize (i.e. tensor over O by O_{𝔭}) the sequence

0 → K → 𝔞^{−1} ⊗_{O} M →^{f} Hom(𝔞, M) → C → 0  (1.28)

and still get an exact sequence (we denote by C = Coker(f) and by K = Ker(f)). The idea is to show that at every localization the map is bijective, so K_{𝔭} and C_{𝔭} are trivial for every 𝔭 ∈ O. A classical result will then imply that K and C are trivial. Take a prime 𝔭 ◁ O, and denote by −_{𝔭} the local version of −.

By the previous lemma, 𝔞_{𝔭} := 𝔞O_{𝔭} is principal for it is invertible. Say 𝔞_{𝔭} = α_{𝔭}O_{𝔭}, then we also have that 𝔞_{𝔭}^{−1} = α_{𝔭}^{−1}O_{𝔭}. Then the localized sequence is

0 → K_{𝔭} → α_{𝔭}^{−1}O_{𝔭} ⊗_{O_{𝔭}} M_{𝔭} → Hom_{O_{𝔭}}(α_{𝔭}, M_{𝔭}) → C_{𝔭} → 0  (1.29)

We begin by showing injectivity. An element u of 𝔞_{𝔭}^{−1} ⊗ M_{𝔭} can be written as

\begin{aligned}u & = ∑ α_{𝔭}^{−1}a_{i}/s_{i} ⊗ m_{i}/t_{i} \\ & = α_{𝔭}^{−1} ⊗ ∑ \frac{a_{i}m_{i}}{s_{i}t_{i}} \end{aligned}  (1.30)

where a_{i} ∈ O, m_{i} ∈ M and s_{i}, t_{i} ∈ O \ 𝔭. Then for x ∈ 𝔞_{𝔭}

f(u)(x) = (α_{𝔭}^{−1}x) ∑ \frac{a_{i}m_{i}}{s_{i}t_{i}}  (1.31)

If f(u)(x) = 0 for all x then it follows that ∑ \frac{a_{i}m_{i}}{s_{i}t_{i}} = 0 for M is torsion-free. Therefore u = α_{𝔭}^{−1} ⊗ 0 = 0

As for surjectivity, let ϕ ∈ Hom_{O_{𝔭}}(𝔞_{𝔭}, M_{𝔭}), then ϕ(α_{𝔭}) = m_{𝔭} determines completely the map as it is a O_{𝔭} homomorphism. So α_{𝔭}^{−1} ⊗_{O_{𝔭}} m_{𝔭} ∈ 𝔞_{𝔭}^{−1} ⊗_{O_{𝔭}} M_{𝔭} is a preimage of ϕ, and the map is surjective.

Replacing the two first elements in the second line of 1.24 by their equivalent module we get

0 → 𝔞^{−1}Λ → 𝔞^{−1}ℂ → Hom_{O}(𝔞, ℂ/Λ) → 0  (1.32)

Thus Hom_{O}(𝔞, ℂ/Λ) can be identified with ℂ/𝔞^{−1}Λ (for 𝔞^{−1}ℂ = ℂ) which is an elliptic curve with CM by O. Interestingly one has a natural map A ≃ ℂ/Λ to 𝔞⋆A ≃ ℂ/𝔞^{−1}Λ of kernel

𝔞^{−1}Λ/Λ = {P ∈ A | aP = O ∀a ∈ 𝔞} =: A[𝔞]  (1.33)

We will now prove a basic fact on invertible ideals in order to get more information this kernel (and on the kernel of [α]_{A}).

Lemma 1.6. Let 𝔞, 𝔟 be integral invertible ideals of O, such that 𝔟 ⊆ 𝔞, then O/𝔟𝔞^{−1} ≃ 𝔞/𝔟.

Proof. Let 𝔟 = k ; ∏_{i=1}𝔭_{i}^{f_{i}}, with f_{i} > 0, and 𝔞 = k ; ∏_{i=1}𝔭_{i}^{e_{i}}, with 0 ≤ e_{i} ≤ f_{i}. Pick x_{i} ∈

𝔭_{i}^{e_{i}} \ 𝔭_{i}^{e_{i}+1}, and let k be the solution of x ≡ x_{i} (mod 𝔭_{i}^{e_{i}+1}) ∀i (which exists by the Chinese remainder theorem). Then the valuation of k ∈ O (well-defined for O_{𝔭_{i}} is a discrete valuation ring) at 𝔭_{i} is e_{i} (i.e. ν_{𝔭_{i}}(k) = e_{i}). We can then define a map m : O/𝔟𝔞^{−1} → 𝔞/𝔟, by sending x ∈ O to kx (well defined as k𝔟𝔞^{−1} ⊆ 𝔟). Localization is an exact functor, so if we prove that the map m is injective and surjective at every prime of O, its kernel and cokernel will be trivial (a O-module is trivial if and only if all its localization are).

Since 𝔞 and 𝔟 are invertible, then every localization is principal (see lemma 1.4) so denote α_{𝔭} and β_{𝔭} as a principal element in 𝔞_{𝔭} and 𝔟_{𝔭} respectively. If 𝔭 = 𝔭_{i} for some i, then k = u_{𝔭}α_{𝔭} where u_{𝔭} is a unit in O_{𝔭} and the multiplication map by α_{𝔭} from O_{𝔭}/(β_{𝔭}α_{𝔭}^{−1}O_{𝔭} to α_{𝔭}O_{𝔭}/β_{𝔭}O_{𝔭} is easily seen to be bijective. If 𝔭 ≠ 𝔭_{i} ∀i then localizing at 𝔭 yields a trivial map: O_{𝔭} = 𝔞O_{𝔭} = 𝔟O_{𝔭}, which is also bijective.

Up to a isomorphism of A we can make 𝔞^{−1}Λ into an integral O-ideal. It then follows that ♯A[𝔞] = N_{K/ℚ}(𝔞). Similarly one can get that ♯Ker([α]_{A}) = N_{K/ℚ}(α).

The action of Pic(O) we defined is simple and transitive. Transitive for if two elements of Pic(O) give the same elliptic curve then they are equal up to homothety in Cl(O):

𝔞 ⋆ A = 𝔟 ⋆ A ⇔ ℂ/𝔞^{−1}Λ ≃ ℂ/𝔟^{−1}Λ
⇔ 𝔞^{−1}Λ is homothetic to 𝔟^{−1}Λ
⇔ 𝔞^{−1} is homothetic to 𝔟^{−1}
⇔ [𝔞^{−1}] = [𝔟^{−1}]

It is simple since the sets Pic(O) and Ell(O) have the same cardinality. As promised, we now inspect the relation between this action and the one from G_{\overline{K}/K}. Theorem 1.7. Let σ ∈ G_{\overline{K}/K}, and 𝔞 ∈ Pic(O) then σ(𝔞 ⋆ A) = 𝔞 ⋆ (σA) Proof. From the isogeny A → A/A[𝔞] one gets ^{σ}A →^{σ} (A/A[𝔞]). Since ^{σ}A also has CM by O, there is another isogeny ^{σ}A → (^{σ}A)/(^{σ}A)[𝔞]. However, by lemma 1.3 their kernel ( ^{σ}(A[𝔞]) and (^{σ}A)[𝔞] ) are equal. But for a given finite subgroup L of E there is an unique elliptic curve and an isogeny f : E → E′ such that Kerf = L (see [22] III.§4). It follows that the images of the two maps are equal and thus that the two action commutes. Remark 1.1. The action of Pic(O) we introduced might seem awkward due to the work involved in proving the commutativity of the actions. Indeed, one could be tempted to define directly 𝔞 ⋆ A = A/A[𝔞]. Though this last definition makes commutativity easy to prove, it does not respects so clearly the properties of an action

(𝔟𝔞 ⋆ A =^{?} 𝔟 ⋆ (𝔞 ⋆ A)).

Since the action of Pic(O) is simply transitive, we can define for a given A ∈ Ell(O) a map η_{A} : G_{\overline{K}/K} → Pic(O) by the rule η_{A}(σ) ⋆ A = 𝔞 ⋆ A. Lemma 1.7. η_{A} is a group homomorphism independent of A. Proof. Independence follows from the commutativity of the actions:

\begin{aligned}η_{A}(^{σ}) ⋆ A′ & = η_{A}(σ) ⋆ 𝔞 ⋆ A = 𝔞 ⋆ η_{A}(σ) ⋆ A \\ & = 𝔞 ⋆ (^{σ}A) =^{σ} (𝔞 ⋆ A) \\ & =^{σ} A′ \end{aligned}  (1.34)

where A′ = 𝔞 ⋆ A And as a simple consequence is a group homomorphism:

η(στ) ⋆ A =^{στ} A =^{σ} (η(τ) ⋆ A) = η(σ) ⋆ η(τ) ⋆ A

Thus we get our first piece of information by looking at

Ker(η) = \{σ ∈ G_{\overline{K}/K} |^{σ} A = A\} = \{σ ∈ G_{\overline{K}/K} |^{σ} j(A) = j(A)\}  (1.35)

for then K(j(A)) = K̄^{Ker(η)} =: H. More importantly G_{H/K} = Im(η) ⊆ Pic(O), and so H is an abelian extension of K.

1.4.4 Explicit Galois Action on j(E)

To get more information about H, we need to use class field theory. As a recall when L/K is an abelian extension of number fields, 𝔭 ⊆ K a prime ideal unramified in L, ℘ a prime of L above (or dividing) 𝔭, l = O_{L}/℘ and k = O_{K}/𝔭, we have an isomorphism between the decomposition group of ℘, D_{℘}, and the galois group G_{l/k}. It is determined by sending the frobenius in G_{l/k} to σ_{℘} ∈ G_{L/K} where the restriction of σ_{℘} to O_{K}/𝔭 corresponds to the frobenius (raising to the |l|^{th} power). Actually σ_{℘} is unique with this property and is independent of ℘, so we will write it σ_{𝔭} instead. The assignment 𝔭 ↦ σ_{𝔭} is actually a isomorphism of Cl(K) with G_{K^{ur}/K} where K^{ur,ab} is the maximal unramified abelian extension of K. When O is not maximal the isomorphism is slightly different: we only consider primes 𝔭 of K prime to the conductor c of O, then

\begin{aligned}rec : & Pic(O) & → & G_{H_{O}/K} \\ & 𝔭 & ↦ & σ_{𝔭O_{K}} \end{aligned}  (1.36)

is an isomorphism, where H_{O} is an abelian extension of K unramified outside the primes dividing c.

Theorem 1.8. H = H_{O}

Proof. Let L = H · H_{O}, then it is an abelian extension of K. Furthermore we have homomorphisms G_{L/K} ↠^{π} G_{H_{O}/K} ≃^{rec} Pic(O), so we define ξ = r ∘ π. On the other hand, since G_{\overline{K}/L} ⊆ Ker(η) we might as well restrict η to G_{L/K}. Then L^{Ker(η)} = H and L^{Ker(ξ)} = H_{O}, so what we really need to prove is that η|_{G_{L/K}}= ξ. To prove this we will first have to describe G_{L/K} properly.

When L/K is a finite extension of number fields, we can define the Tchebotarev density δS of a set S of primes of L. The properties of this density that are of interest to us are the following:

1. If T ⊆ S is a finite set then δS = δ(S \ T).

2. If T = {℘ in S with N_{L/ℚ}(℘) a rational prime} then δS = δ(T).

3. If δS = 1 then G_{L/K} = {σ_{𝔭} | ∃℘ ∈ S such that 𝔭 divides ℘}.

Note that since every ℘ ∈ L divides some 𝔭 ∈ K, and that σ_{℘} = σ_{𝔭} by definition, it is more convenient to speak of the set S as a set of primes of K. The set S that we will use is define as all the primes 𝔭 of O satisfying:

C1− 𝔭 is prime to c the conductor of O.

C2− 𝔭 is unramified in L.

C3− Every elliptic curve A ∈ Ell(O) has good reduction at all ℘ | 𝔭.

C4− N_{K/ℚ}(𝔭) is a rational prime.

C5− 𝔭 does not divide N_{L/K}(j(A) − j(A′))O_{K} ∀A, A′ ∈ Ell(O) such that A ≠ A′. So our set S has density 1 (at every step except C4 we took out a finite number of

prime, and C4 is a condition that does not affect δS). The purpose of these conditions will be made clear as we go trough the proof. According to the third property of the density, and using our first condition on S, one notes that we now only need to show that η(σ_{𝔭}) = ξ(σ_{𝔭}) = [𝔭] to have η = ξ.

Let β ∈ 𝔭 \ 𝔭^{2} and 𝔮 be such that 𝔭𝔮 = pO for p ∈ ℤ (exists by C1, C2 and C4). Then, by the Chinese remainder theorem, there is a α ∈ O such that

α ≡ β (mod 𝔭^{2}), α ≡ 1 (mod 𝔮), α ≡ 1 (mod c)  (1.37)

Consequently αO = 𝔞𝔭 for some 𝔞 ◁ O such that 𝔞 is prime to both c and 𝔭. For some A ∈ Ell(O) consider the maps

A →^{φ} 𝔭 ⋆ A →^{ψ} 𝔞𝔭 ⋆ A = (α) ⋆ A ≃^{μ} A  (1.38)

Then μ ∘ ψ ∘ φ = [α]_{A}, so in particular the isogeny is of degree N_{K/ℚ}(α) and (μ ∘ ψ ∘ φ)^{∗}ω = αω. For some ℘ | 𝔭, we now reduce everything modulo ℘ | 𝔭 (note that no 2 j-invariants become equal due to C5), and denote by ˜−_{℘} the reduced version of −. Then ˜[α]_{℘} = μ̃_{℘} ∘ ψ̃_{℘} ∘ φ̃_{℘} is inseperable since ℘|α (α ∈ 𝔭) and ˜[α]^{∗}_{℘}ω̃_{℘} = α̃_{℘}ω̃_{℘}. The degree of the isogenies is not changed by good reduction (C3) and α ∈ 𝔭. Furthermore, deg(μ̃_{℘}) = deg(μ) = 1 since μ is an isomorphism, and deg(ψ̃_{℘}) = deg(ψ) = N_{K/ℚ}(𝔞) is separable since 𝔞 is coprime to 𝔭. Thus, φ̃_{℘} is of degree p (C4)and must be inseparable. It can be expressed as F r ∘ λ, where F r is the p^{th}-power Frobenius map and λ is separable. Since the Frobenius is of degree at least p it follows that λ is of degree one, so an isomorphism. Consequently, 𝔭 ⋆ Ã_{℘} = φ̃_{℘}(Ã_{℘}) = F r(Ã_{℘}) = Ã_{℘}^{(p)}, where Ã_{℘}^{(p)} is the curve Ã_{℘} with all the coefficients to the p^{th} power. By definition of j, we have j(Ã_{℘}^{(p)}) = j(Ã_{℘})^{p}, so

j(𝔭 ⋆ A) = j({ \atop 𝔭 ⋆ A}_{℘}) = j(Ã_{℘}^{(p)}) = j(Ã_{℘})^{p} ≡ σ_{𝔭}j(Ã_{℘}) ≡ j(σ_{𝔭}Ã_{℘}) (mod ℘) (1.39)

But then

j(𝔭 ⋆ A) ≡ j(σ_{𝔭}A) (mod ℘)  (1.40)

So since the j-invariants stay different (C5), we have that

\begin{aligned}& j(E) ≡ j(E′) (mod ℘) ⇔ E ≃ E′ \\ ⇒ & 𝔭 ⋆ A ≃ σ_{𝔭}A = η(σ_{𝔭}) ⋆ A \end{aligned}  (1.41)

Thus ξ(σ_{𝔭}) = [𝔭] = η(σ_{𝔭}), η = ξ and H = H_{O}

In fact we have proved much more: we have an explicit relation between the action of G_{H_{O}/K} with that of Pic(O) by

σ_{𝔭}j(A) = j(𝔭 ∗ A)  (1.42)

Now if we let Λ_{A} be the lattice associated to A, then Λ_{A} ≃ ⟨1, τ⟩ with ℑ(τ) > 0 , then we define j(A) = j(τ). Since Λ_{𝔭⋆A} = 𝔭^{−1}Λ_{A} we will write

σ_{𝔭}j(τ) = j(𝔭 ⋆ τ),  (1.43)

where ⟨1, 𝔭 ⋆ τ⟩ ≃ 𝔭^{−1}Λ_{A}.

Furthermore, since ⟨1, τ⟩ homothetic to ⟨cτ +d, aτ +b⟩ if ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ), one sees that τ is only well-defined up to an action by SL_{2}(ℤ) defined by

(\begin{matrix}a & b \\ c & d \end{matrix}) τ = \frac{aτ + b}{cτ + d}  (1.44)

These observations enables us to rewrite Ell(O) as the set

{τ ∈ SL_{2}(ℤ)\H | O_{⟨1,τ⟩} = O}  (1.45)

where H = {z ∈ ℂ | ℑ(τ) > 0} and using the usual equivalence between latices and elliptic curves.

26  Elliptic Curves

Chapter 2

Heegner Points

A crucial tool in the construction of Heegner points is the modular parametrization. We will briefly describe the properties of this parametrization and the objects that are necessary to define it. We will follow by the definition of Heegner points, their properties, how to compute them, and a quick survey of the results of our computations on curves of conductor less than 3000.

2.1 Modular Parametrization

Through the work of many mathematicians there emerged the remarkable conjecture that elliptic curves over ℚ could be in some sense parametrized by other objects, namely cusp forms. This insight has now been confirmed to be true. Consequently, we will devote this first section to a very short outline of the results that will be needed to make proper definitions of Heegner points and their properties.

There are many reasons to consider the upper-half plane H = {z ∈ ℂ | ℑ(z) > 0} and the action of SL_{2}(ℤ) that one defines on it. One of them is introduced in the appendix, where (quadratic) points in the upper half plane represent a positive definite primitive binary quadratic form, and another one comes from the description of representatives for Ell(O), as seen in 1.44. These notions of equivalence then yield the action of SL_{2}(ℤ) on H, and we can obtain a fundamental domain F for SL_{2}(ℤ)\H (see A.5).

On the other hand H is also a Riemann surface, and the group of matrices of positive determinant GL_{2}^{+}(ℝ) acts on H as in A.4 or 1.44. Since scalar multiples of the identity act trivially on H, we can consider this as the action of the group PGL_{2}^{+}(ℝ) = GL_{2}^{+}(ℝ)/ℝ^{×}I. Furthermore, this action leave both the hyperbolic line element ds^{2} = dz dz̄/y^{2} = (dx^{2} + dy^{2})/y^{2} and the hyperbolic surface element dμ = dz ∧ dz̄/2iy^{2} = dx ∧ dy/y^{2} unchanged (here x + iy = z ∈ H). Indeed for γ = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}^{+}(ℝ), one has

d(γz) = \frac{dγz}{dz}dz
= ((cz + d)a − (az + b)c)h(γ, z)^{−2}dz
= Det(γ)h(γ, z)^{−2}dz

where h(γ, z) = cz + d, and similarly d(γz̄) = Det(γ)h(γ, z̄)^{−2}dz̄. Since ℑ(γz) = y|h(γ, z)|^{−2} it follows that the action of GL_{2}^{+}(ℝ) is a hyperbolic isometry. Since the riemannian structure is invariant under these transformations, it can also be defined on the quotient of H by any subgroup of GL_{2}^{+}(ℝ). In fact, for some subgroups, this quotient is also a Riemann surface. The subgroups that are of interest to us are the Hecke congruence subgroups:

Γ_{0}(N) = {γ ∈ SL_{2}(ℤ)| γ is upper triangular (mod N)}

The quotients Γ_{0}(N)\H are not compact (see for example the description of SL_{2}(ℤ)\H in A.5). This can be corrected by considering H^{∗} = H ∪ ℚ ∪ {∞}. We call the points added to H the cusps. One can check that the action of SL_{2}(ℤ) extends properly to these points given that ( \begin{matrix}a & b \\ c & d \end{matrix}) ∞ = a/c. To make H^{∗} a topological space, we define the neighborhoods of ∞ to be the points of imaginary value greater than a given constant. For compatibility, their image under SL_{2}(ℤ) gives the neighborhood of points in ℚ.

It can be shown (see [15] Ch.XI or [19] Ch.II) that the quotients X_{0}(N) := Γ_{0}(N)\H^{∗} can be given the structure of a compact Riemannian surface.

Remark that since T = \{( \begin{matrix}1 & k \\ 0 & 1 \end{matrix} ) |k ∈ ℤ\} forms an infinite subgroup of SL_{2}(ℤ), every finite index subgroup of SL_{2}(ℤ) possess a non-trivial subgroup of T. It is useful to call the width of a cusp x = α^{−1}∞ the positive integer k_{α} such that ( \begin{matrix}1 & k_{α} \\ 0 & 1 \end{matrix} ) generates T ∩ αΓ_{0}(N)α^{−1}. It does not depend on the matrix α, but only on the cusp x.

To study a compact Riemannian surface, it is natural to consider its holomorphic differentials. In the case of X_{0}(N) they can be seen as very explicit objects namely cusp forms of weight 2 for Γ_{0}(N). For α ∈ GL_{2}^{+}(ℝ), denote by f |_{α}(z) := Det(α)h(α, z)^{−2}f(αz).

Definition 2.1. A cusp form of weight 2 for Γ_{0}(N) is a function f : H → ℂ such that

1. f(z) = f(γz) h(γ, z)^{−2} for all γ ∈ Γ_{0}(N).

2. For all α ∈ SL_{2}(ℤ), the function f |_{α}(z) = h(α, z)^{−2}f(αz) possesses a Fourier

expansion of the form

f |_{α}= ∑_{n>0}a_{n}^{(α)}e^{2πinz/k_{α}}  (2.1)

In particular, since ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ∈ Γ_{0}(N), a cusp forms f can be written as ∑ a_{n}q^{n} with q = e^{2πiτ} and a_{1} = a_{1}^{∞}. Cusp forms form a vector space over ℂ denoted by S_{2}(N). To describe the relation that these objects have with elliptic curve we still need to introduce more notations.

First, note that if f ∈ S_{2}(N) then, for s ≥ 1, f is also in S_{2}(sN) . Furthermore, if α_{s} = ( \begin{matrix}1 & 0 \\ 0 & s \end{matrix}) and g(z) = f |_{α_{s}} (z) = s^{−1}f(s^{−1}z) then g ∈ S_{2}(sN). Indeed, since α_{s}^{−1}Γ_{0}(sN)α_{s} ⊆ Γ_{0}(N), then f |_{α_{s}}|_{γ}= f |_{α_{s}^{−1}γα_{s}}|_{α_{s}}= f |_{α_{s}} for any γ ∈ Γ_{0}(sN). Cusp forms in S_{2}(N) obtained from cusp forms in S_{2}(N′) for some N′ | N and N′ ≠ N, are called old forms, and the space they generate is denoted S_{2}^{old}(N). These are the elements that are not of interest to us, and in order to define a complementary space we need some extra structure on the vector space. Definition 2.2. The Peterson scalar product for f, g ∈ S_{2}(N) is

⟨f, g⟩ = ∫_{H/Γ_{0}(N)} f(z)\overline{g(z)}dx dy  (2.2)

The region Γ_{0}(N)\H is described as follows: we write SL_{2}(ℤ) = ∪_{i}β_{i}Γ_{0}(N) and obtain a fundamental domain for Γ_{0}(N)\H as ∪β_{i}^{−1}F, where F is the fundamental domain of SL_{2}(ℤ)\H. It is not too hard to check that the integral is convergent (due to the behavior at the cusp) and that it is well-defined (i.e. that it does not depend on the choice of the fundamental region). It is also non-degenerate (see [20]).

It is now possible to define a perpendicular subspace to S_{2}^{old}(N). It is denoted S_{2}^{new}(N). One is then led to consider a family of self-adjoint operators (Hecke operators see [15]) for this new space. Since these operators commute, it is possible to find a basis of simultaneous eigenvectors of S_{2}(N). We say f is a newform of level N when it is a simultaneous eigenvector and its first Fourier coefficient a_{1} is 1. It is actually possible to choose a basis for S_{2}(N) consisting of newforms with integer Fourier coefficients.

To a newform of level N we attach an L-function by setting

L(f, s) = ∞ ; ∑_{n=1}a_{n}n^{−s}  (2.3)

This function satisfies many of the properties that the one defined for elliptic curves does. Theorem 2.1. (Eichler-Shimura construction) Given f a newform of level N with integer Fourier coefficients there exists an elliptic curve E_{f} over ℚ such that L(E_{f}, s) = L(f, s)

For a discussion of this construction see [6], [15] or [21]. An important step in the proof is to realize that X_{0}(N) is an algebraic curve over ℂ that possess a model over ℚ. Simply said it is in bijection with a curve described by the vanishing of a polynomial with coefficients in ℚ. In this model, points on the modular curve X_{0}(N) are given by a pair of elliptic curves (actually, their j-invariant) related together by a cyclic N-isogeny (an isogeny whose kernel is cyclic and of cardinality N)

Crucial to our aims is the fact that this construction also gives an algebraic map Φ_{N} : X_{0}(N) → E_{f} where E_{f} is obtained by the Eichler-Shimura construction. Further, one has that Φ_{N}^{∗}(ω) = c2πif(τ)dτ where ω is the Néron differential of E_{f}, c ∈ ℚ^{∗} is a constant (called the Manin constant of E_{f}), and f(τ)dτ is the differential of X_{0}(N) given by f. This map can be actually be made explicit:

Theorem 2.2. Let Λ_{E} be the Néron lattice of E_{f} and let c be the Manin constant attached to E_{f}. Let Φ_{w} : ℂ/Λ_{E} → E(ℂ) be the Weierstrass uniformisation. Then for τ ∈ H^{∗},

Φ_{N}(τ) = Φ_{w}(z_{τ}) where z_{τ} = c ∫_{i∞}^{τ} 2πif(z)dz = c ∞ ; ∑_{n=1} \frac{a_{n}}{n} q^{n}, with q = e^{2πiτ} (2.4)

Though the description given here is purely analytic, this is actually an algebraic map between X_{0}(N) and E_{f}. It can also be showed that this maps sends cusps to torsion points (see [15] XI). One question naturally raised from this construction is which curves can be obtained in such a manner. It has been recently solved thanks to a fundamental breakthrough of Wiles. Indeed, every isogeny class possesses an elliptic curve which can be constructed from a newform, and this specific curve is called the strong Weil curve. If a curve is not a strong Weil curve then we say it is a weak Weil curve in the isogeny class.

2.2 Definition

Considering that we have an algebraic map from X_{0}(N) to an elliptic curve E of conductor N, and that our aim is to produce a point in E defined over ℚ, it seems reasonable to look at points on X_{0}(N) on which we have some Galois theoretic control. The definition then comes naturally (see also [10]):

Definition 2.3. A Heegner point (on X_{0}(N)) associated to the order O (of a quadratic imaginary field K) is a point (j(A), j(A′)) where A, A′ both have complex multiplication by O and are, like all points of X_{0}(N), related by a cyclic N-isogeny.

Alternatively, using Weierstrass description of elliptic curves over ℂ, we can see the isogeny as a map A ≃ ℂ/Λ → ℂ/Λ′ ≃ A′ where Λ and Λ′ are the lattices associated to A and A′. Since the isogeny is cyclic and we can use a homothety of ℂ^{×} (i.e. an isomorphism over ℂ), to have that Λ ⊆ Λ′ ⊆ O and Λ′/Λ ≃ ℤ/N. Then 𝔫 = Λ · Λ^{′−1} is an integral ideal. A consequence of interest to lemma 1.6 is that we can get a nice description of 𝔫:

O/𝔫 = O/Λ · Λ^{′−1} ≃ Λ′/Λ ≃ ℤ/N  (2.5)

It follows that 𝔫 is of norm N_{K/ℚ}(𝔫) = N and yields a cyclic quotient.

The first question that one should ask is whether or not it is possible choose such a pair of elliptic curves. The above discussion already gives us part of the answer:

Theorem 2.3. For a given an order O of discriminant D, and a conductor N ∈ ℤ, the following are equivalent:

1. A Heegner point exists on X_{0}(N).

2. There exists an ideal 𝔫 of O such that 𝔫 is cyclic of norm N, i.e. O/𝔫 = ℤ/N.

3. The equation D = B^{2} − 4NC has a solution (B, C) with gcd(N, B, C) = 1.

Proof. The fact that 1 ⇔ 2 follows from discussion we just made: we have that given a Heegner point we can produce the required ideal as 𝔫 = Λ · Λ^{′−1}. Conversely given an ideal 𝔫 ◁ O, then any invertible ideal 𝔞 ◁ O will be a lattice, thus yields an elliptic curve. Taking Λ′ = 𝔞 · 𝔫^{−1} and Λ = 𝔞 we have the required pair of elliptic curves.

The proof of 2 ⇔ 3 comes from the theory of binary quadratic form, thus it has been relegated to appendix A, lemma A.3.

Remark 2.1. It is important to point out that this theorem also gives us a complete description of a Heegner point as a triplet (O, 𝔫, 𝔞), since the last two elements give all the information that is contained in the pair (A, A′) and the cyclic N-isogeny that relates them, and O is their endomorphism ring. Furthermore, as the curve ℂ/𝔞 is defined up to isomorphism, the ideal 𝔞 is only defined up to principal ideals (homothety of lattices). As a consequence we will write Heegner points as (O, 𝔫, [𝔞]) where [𝔞] is the equivalence class of 𝔞 in the class group Cl(O). The conductor of the order O is often referred to as the conductor of the Heegner point.

Let c be the conductor of O and suppose that gcd(c, N) = 1, then condition 3 in theorem 3.2 is just D ≡ B^{2} (mod 4N). Indeed, write D = c^{2} · d (where d is a fundamental discriminant) and suppose that D = B^{2} − 4NC. If there is a p| gcd(N, B, C) then p^{2} | d which is impossible unless p = 2. If this is the case, then one notes that d ≡ 8 or 12 (mod 16) and B^{2} ≡ 0 or 4 (mod 16) a contradiction.

Definition 2.4. We say that an order O ⊆ K of discriminant D and conductor c satisfies the Heegner Hypothesis (HH) for a given elliptic curve of conductor N if D is a square mod4N and gcd(c, N) = 1.

This definition is actually slightly weaker than the usual one, in which we ask for the discriminant to be coprime to N. We will refer to the latter as the Strong Heegner Hypothesis (SHH). Note that some cases fall outside HH, for example, a curve of conductor N = 40 with D = −96 has a cyclic ideal of norm 40, namely ⟨40, 4 + 2\sqrt{−6}⟩. Remark that this is not an invertible ideal, and is consequently very awkward to work with.

2.3 Galois action on Heegner points

Now that we have defined a proper point on X_{0}(N), we can start to apply the tools that come from CM theory. As a recall, we had a curve E′ with CM by O, and we defined τ such that ⟨1, τ⟩ is homothetic to the lattice associated to E′. We then had to consider O = O_{⟨1,τ⟩}, so that ∀α ∈ Cl(O), j(α ⋆ τ) = rec(α)^{−1}j(τ). An important point in the proof of this result is that j(τ) belongs to the Hilbert class field of O, H_{O}.

Since we will want to apply this theorem to the point (j(A), j(A′)) on X_{0}(N) we first have to describe the action on the two j-invariants. First, let’s define υ and τ such that Λ ≃ ⟨1, υ⟩ and Λ′ ≃ ⟨1, τ⟩. Note that we can choose them so that υ = Nτ, as there is a cyclic N-isogeny A → A′, to avoid the need of specifying both of them. We are then led to consider the order O_{τ,N} = O_{⟨1,Nτ⟩} ∩ O_{⟨1,τ⟩} as its class field H_{O_{τ,N}} contains both j(τ) and j(Nτ). This can be seen using the description 1.43 of the action of the Galois group of H_{O_{τ,N}}. Thus the Heegner point X_{τ} = (j(Nτ), j(τ)) is in X_{0}(N)(H_{O_{τ,N}}). The following corollary follows directly since the map Φ_{N} : X_{0}(N) → E is a map of algebraic curves defined over ℚ. Corollary 2.1. The image of the Heegner point X_{τ} by Φ_{N} lies in E(H_{O_{τ,N}}).

So far, we have found a point on E in a specific number field. As we wish to obtain a point on E(ℚ) from this, we will now look at the Galois action. Using the reciprocity map, this will translate to the familiar action of Cl(O_{τ,N}):

rec(𝔞)^{−1}(j(Nτ), j(τ)) = (rec(𝔞)^{−1}j(Nτ), rec(𝔞)^{−1}j(τ))
= (j(𝔞 ⋆ Nτ), j(𝔞 ⋆ τ))
= (j(υ′), j(τ′))

(2.6)

Where υ′ and τ′ are both defined modulo the action of SL_{2}(ℤ). However there is still a cyclic N-isogeny between the elliptic curves represented by υ′ and τ′ thus they can again be chosen so that υ′ = Nτ′.

It is convenient to write this action as 𝔞 ⋆_{N} τ = τ′ to emphasize it is a level N lift of the ⋆ action. Also note that the result τ′ is defined up to Γ_{0}(N). Indeed, if (( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(N) then ⟨1, ( \begin{matrix}a & b \\ c & d \end{matrix}) τ⟩ and ⟨1, N ( \begin{matrix}a & b \\ c & d \end{matrix}) τ⟩ are both homothetic (by a factor of cτ + d) to ⟨1, τ⟩ and ⟨1, Nτ⟩. It then makes sense to define the set of Heegner points as HP(O) = {τ ∈ H/Γ_{0}(N) | O_{τ,N} = O}. In fact, written like this, the elements in HP(O) are just different representatives of the elements in Ell(O). We will now make explicit the relation between this description and the previous one (O, 𝔫, [𝔞]), and doing so will enable us to get a very simple method to determine the Galois action and the Heegner points.

It is actually very easy to state the actual relation between these two notations. From 𝔫 and 𝔞 we produce the lattice Λ, and then get a homothetic lattice whose generators are 1 and τ. However the choice of τ is not fortuitous, since we must make sure that O_{τ,N} = O.

O_{τ,N} = O_{⟨1,Nτ⟩} ∩ O_{⟨1,τ⟩}

= {α ∈ O|α · ⟨1, Nτ⟩ ⊆ ⟨1, Nτ⟩ and α · ⟨1, τ⟩ ⊆ ⟨1, τ⟩}

= \{α ∈ O|∃ ( \begin{matrix}a & b \\ c & d \end{matrix}) , ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ M_{2}(ℤ) such that α\binom{Nτ}{1} = ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) \binom{Nτ}{1}

and α\binom{τ}{1} = ( \begin{matrix}a & b \\ c & d \end{matrix}) \binom{τ}{1}\}

= \{α ∈ O|∃ ( \begin{matrix}a & b \\ c & d \end{matrix}) , ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ M_{2}(ℤ) such that α\binom{τ}{1} = ( \begin{matrix}1/N & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ( \begin{matrix}N & 0 \\ 0 & 1 \end{matrix} ) \binom{τ}{1}

and α\binom{τ}{1} = ( \begin{matrix}a & b \\ c & d \end{matrix}) \binom{τ}{1}\}

= \{α ∈ O|∃ ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ M_{2}(ℤ) and N |c such that α\binom{τ}{1} = ( \begin{matrix}a & b \\ c & d \end{matrix}) \binom{τ}{1}\}

= \{( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ M_{2}(ℤ) and N |c such that τ = ( \begin{matrix}a & b \\ c & d \end{matrix}) τ := \frac{aτ+b}{cτ+d}\}

(2.7)

It is convenient to define M_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ M_{2}(ℤ) and N | c\}, the set of upper triangular matrices modulo N. In the last step we replace the usual quadratic imaginary order by an order of matrices. Note that one can pass from one to the other by looking at the eigenvalues of such matrices. Their eigenvectors are \binom{τ}{1} and \binom{τ̄}{1}, and by taking the eigenvalue associated to the first one , we recover an element of O_{τ,N}.

From this one can explicitly describe a Heegner point. Suppose O = ⟨1, ω_{D}⟩. We will use 2.7 to find a τ such that O_{N,τ} = O. Since the identity matrix is always in O_{N,τ}, it suffices to find another one A = ( \begin{matrix}a & b \\ c & d \end{matrix}) with N | c that behaves like ω, i.e. satisfies A^{2} − T r_{K/ℚ}(ω_{D}) · A + N_{K/ℚ}(ω_{D}) = 0. From small calculations one finds that this is equivalent to T r(A) = T r_{K/ℚ}(ω_{D}) and Det(A) = N_{K/ℚ}(ω_{D}). τ can finally be obtained by solving τ = Aτ so τ = ((2a − T r_{K/ℚ}(ω)) + \sqrt{D})/2c where D is the discriminant of O. Interestingly (2a − T r_{K/ℚ}(ω_{D}))^{2} = −4bc + D ≡ D (mod 4N). Thus s = 2a − T r_{K/ℚ}(ω_{D}) is a square root of D modulo 4N.

Though the preceding method is very efficient to compute one Heegner point, and does not rely on the more cumbersome (O, 𝔫, [𝔞]) notation, it does not help expressing the Galois action on such points. For this, it is more useful to proceed using binary quadratic forms. As in the appendix, here I will denote the map that sends a form f to its associated integral ideal.

To get τ from (O, 𝔫, [𝔞]) one has to proceed with more care than simply picking a point in H with ⟨1, τ⟩ ≃ 𝔞𝔫^{−1}. For a start, ⟨1, τ⟩ must have endomorphism ring O; it is a fractional O-ideal. As such, it is associated to a primitive binary quadratic form f of discriminant D = Disc(O) (i.e. f(τ, 1) = 0). Indeed, if f(x, y) = Ax^{2} + Bxy + Cy^{2} is such that τ_{f} = τ, we must have that N | A and B^{2} ≡ D (mod 4N), in order to have O_{τ,N} = O. Furthermore, as ⟨1, τ⟩ ≃ Λ′, I(f) ∈ [Λ′] = [𝔞𝔫^{−1}]. The ideal 𝔫^{−1} is in turn represented by a form g(x, y) = Nx^{2} − sxy + jy^{2} where s is a square root of D mod 4N and 4Nj = s^{2} − D (see lemmas A.2 and A.3, note the −s as opposed to s when we are looking at 𝔫). Let h(x, y) be the form associated to some 𝔦 ∈ [𝔞], then Λ′ = 𝔦𝔫^{−1} is equivalent to f = g · h. Finally, to have that N | A and B^{2} ≡ D (mod 4N) (so B is determined mod2N), it follows that 𝔦 must be chosen so that the Dirichlet composition of g and h is possible (it always exists by remark A.1). So τ is obtained as the point in H associated to g · h where I(g) = 𝔫^{−1}, and I(h) = 𝔦 (𝔦 chosen as above).

We work out the inverse correspondence as follows. First, from τ we recover O = O_{τ,N}. Second, we find the binary quadratic form of τ = \frac{s+\sqrt{D}}{2kN} ∼ f(x, y) = kN x^{2} + s xy + j y^{2}. Then we define 𝔫^{−1} as the ideal represented by g(x, y) = Nx^{2} + (s mod 2N)xy + j′y^{2}, and 𝔦 ∼ h(x, y) = kx^{2} + (s mod 2k)xy + j′′y^{2}. Since gcd(N, k, s) = 1, Dirichlet composition of g and h is possible and f = g · h ⇒ ⟨1, τ⟩ ≃ 𝔦𝔫^{−1}. Consequently (O, 𝔫, [𝔦]) is just the long notation for τ.

Theorem 2.4. Let (O, 𝔫, [𝔞]) be a Heegner point, and 𝔟 ∈ Cl(O) then

rec(𝔟)^{−1}(O, 𝔫, [𝔞]) = (O, 𝔫, [𝔞𝔟^{−1}])  (2.8)

Proof. This result is direct from the translation of the two notations and the definition of ⋆_{N}. Indeed, 𝔫^{−1}[𝔞] is related to τ by the usual homothety. On the other hand, τ′ = 𝔟 ⋆_{N} τ is defined so that ⟨1, τ′⟩ = ⟨1, τ⟩I^{−1} where I is a representative of [𝔟]. τ′ then translates back to a representative of 𝔫^{−1}[𝔞𝔟^{−1}].

As a consequence of this, one can note that the ⋆_{N} action on HP(O) is simply transitive. We are now also able to produce a point on E(ℚ): first we can make a point P_{K} ∈ E(K) simply by adding all the points in Φ_{N}(HP(O)), and then P_{K} + \overline{P_{K}} yields the desired point. Unfortunately, we have no guarantee that this point is not the point at infinity.

However, getting a point on E(ℚ) is almost nothing when compared to the properties of P_{K}. Gross and Zagier [11] made an impressive link between this a priori algebraic data and the first derivative of the L-function of E(K). They proved that ⟨P_{K}, P_{K}⟩, where ⟨ , ⟩ denotes the Néron-Tate height on E(K), is proportional to L′(E/K, 1), and as a consequence, P_{K} is torsion ⇔ L′(E/K, 1) = 0. Kolyvagin ([16] and [17]) then showed that we can get an upper bound on the Mordell-Weil group E(K). Indeed, if P_{K} is non-torsion, then it generates a finite-index subgroup of E(K). However, this also shows that P_{K} cannot be properly used on curves with rank bigger than one. Combining these two important facts about P_{K}, it is possible to prove that if the analytic rank of the curve E(ℚ) is 0 or 1, then it is equal to the algebraic rank of E.

2.4 Calculation of Heegner Points

One of the most exceptional properties of Heegner points is that they are very simple to compute as was sketched in the preceding section. Indeed, they are described only in terms of binary quadratic forms, and by virtue of theorem 2.2, we can explicit the point to which they map to. In order to illustrate this, we will produce some examples, beginning trivial class group examples before moving to slightly more complex cases.

Example 2.1. Let’s begin by the first curve on Cremona’s table (see [5]): by^{2} + y = x^{3} − x^{2} − 10x − 20 of conductor 11. First we must find a D (discriminant of a quadratic field) which is a square modulo 4 · N = 44. A nice choice would be −43, as its class group is trivial and its square root mod4N is 1. So ♯HP(O) = 1, and we can get τ by the first method mentioned: −43 = 1^{2} (mod 44) ⇒ s = 1, and A = ( \begin{matrix}1 & 11 \\ 1 & 0 \end{matrix} ) ⇒ τ = (1 + \sqrt{−43})/22. The z_{τ} in theorem 2.2 is then given by

∑_{n≥1} a_{n} · q^{n}/n = q − q^{2} − q^{3}/3 + …  (2.9)

(the Manin constant for the strong Weil curves of conductor less than 8000 has been found to be one) where q = e^{2πiτ}. The result of this sum maps to a point on the elliptic curve using Weierstrass uniformisation: Φ_{w}(z) = (53 − 4\sqrt{−43}, −347 + 44\sqrt{−43}). Of course, we have not proved that this is the image, but it numerically fits to a few thousands of digits, and is an exact solution to the Weierstrass equation.

Example 2.2. Let’s take the first curve of rank one that appears in the tables: y^{2} + y = x^{3} − x of conductor 37. If we pick -67 (≡ 9^{2} (mod 148)) as our discriminant for the quadratic field, we again conveniently find ourself in a trivial class group case, thus we need only to find one τ = (9 + \sqrt{−67})/74. Summing up the a_{n}e^{2πniτ}/n and using Φ_{w} we are given the point (6, −15). One can check that this point is of infinite order.

When the class group is non-trivial, it becomes more convenient to see Heegner points as the triplet (O, 𝔫, [𝔞]). If for each equivalence class of Cl(O), we take a representative say 𝔞_{i} for i = 1, …, h(O), we can associate to each of them a distinct τ_{i} ∈ HP(O) as described in the previous section. For these computations, one can rely only on binary quadratic forms. We first compute the class group h_{i}(x, y) as in Remark A.2, and find the form g(x, y) associated to our ideal of norm N. If h_{i} cannot be composed with g using Dirichlet composition, then we replace it by an equivalent form for which it is. Once we have a proper set of representatives for Cl(O), we have our τ_{i} ∼ f_{i}(x, y) = g(x, y) · h_{i}(x, y). Example 2.3. The first curve for which no O with Cl(O) = 1 satisfies the SHH is y^{2} + xy + y = x^{3} + 4x − 6 of conductor 14. We can take D = −31 ≡ 5^{2} (mod 56), which gives

\begin{aligned}h_{1}(x, y) & = x^{2} + xy + 8y^{2}, \\ h_{2}(x, y) & = 2x^{2} − xy + 4y^{2}, \\ h_{3}(x, y) & = 2x^{2} + xy + 4y^{2}, \\ g(x, y) & = 14x^{2} + 5xy + y^{2} \end{aligned}  (2.10)

Only h_{2} does not satisfy our requirement (gcd(14, 2, (5 − 1)/2) = 2), but it is equivalent to 4x^{2} + xy + 2y^{2} that does. As a consequence

\begin{aligned}f_{1}(x, y) & = 14x^{2} + 5xy + y^{2} & ⇒ τ_{1} = (−5 + \sqrt{−31})/28 \\ f_{2}(x, y) & = 56x^{2} + 33xy + 5y^{2} & ⇒ τ_{2} = (−33 + \sqrt{−31})/112 \\ f_{3}(x, y) & = 28x^{2} + 33xy + 10y^{2} & ⇒ τ_{3} = (−33 + \sqrt{−31})/56 \end{aligned}  (2.11)

We can then find the 3 points to which they correspond on E(H_{O}), however their algebraic expression can be very complicated. It is easier to look at the polynomial their x and y coordinates satisfy:

X^{3} + X^{2} (\frac{5 − 5\sqrt{−31}}{2}) + X(−27 + 2\sqrt{−31}) + (12 − 3\sqrt{−31}) for the x-coordinate

Y^{3} + Y^{2} (\frac{−85 + 5\sqrt{−31}}{2}) + Y (35 + 48\sqrt{−31}) + (404 + 11\sqrt{−31}) for the y-coordinate (2.12)

or to compute the point P_{K} = ∑ Φ_{N}(τ_{i}) = ((\frac{1−\sqrt{−31}}{2}) , \sqrt{−31})

An interesting thing to point out is that the points we obtained so far on the elliptic curves have coordinates which are algebraic integers, and none of them are torsion points. One can ask whether this is always the case. The answer is no. For example the first strong Weil curve appearing in the tables that has non-integral Heegner points is 33A1 (with D = −8 for example); note that 11A2 is actually the first curve in the tables not to possess this property. Also, if one looks at the curve 121A1 (y^{2} + xy + y = x^{3} + x^{2} + x + 1) and pick D = −19, then τ = \frac{49+\sqrt{−19}}{242} and Φ_{N}(τ) is the point at infinity.

2.5 Numerical Results

We will now comment on the Heegner points that were calculated on all the elliptic curves of conductor less than 3000. The actual data being too lengthy to fit in these pages, it is going to be available on Henri Darmon’s website.

2.5.1 Modular uniformisation

First there is an interesting application of Heegner points to the Manin constant c. Here are a few facts (see [1], [8], [9] and [18]) that we know about it:

1 − c is an integer

2 − if p | c then p = 2, 3, 5 or 7 3 − if p | c then p | N 4 − if p | c then p^{2} | 4N 5 − if 4 | c then 4 | N

So we can only be certain that c = 1 when N is odd and not divisible by 9, 25, and 49. It has been verified by Cremona that the strong Weil-curve of each isogeny class (the first one to appear in the tables) has c = 1. Since the points z_{τ} are multiplied by c before using Weierstrass uniformisation, it could be possible that assuming c = 1 yield points that are not algebraic. In fact it happens for the curve 27A3 (y^{2} + xy = x^{3}), where c = ±3 or ±9 give algebraic points but not c = 1 (at least not the first thousand digits). Similarly for 32A2 (y^{2} = x^{3} − x), and this time c = ±2 or ±4 give algebraic points.

2.5.2 Integrality

As was mentioned in the preceding section, on some curves the Heegner points are algebraic integers for a given discriminant. Surprisingly, on some curves these points seem to be integral for any given discriminant (the first example is 11A1 on Cremona’s table, for which it has been tested for a hundred discriminants). Sometimes, the minimal polynomial satisfied by the Heegner points is even defined over the order we chose to get the points (which is not necessarily maximal), and again some curves seem to possess this property for any discriminant (the first example is 37A1).

Definition 2.5. (IP) An elliptic curve E of conductor N is said to have the integrality property if for all (not necessarily fundamental) discriminant D < 0 with D ≡ B^{2} (mod 4N) for some B ∈ ℤ, the Heegner points are integral points of E over the Hilbert class field of O_{D}, i.e. their coordinates belong to O_{H_{D}}. (SIP) If moreover the coordinates of the Heegner points satisfy polynomials in O_{D}[x], then we say the curve have the strong integrality property.

Of course, squares mod4N coprime to N are infinite, since such numbers need only give satisfies congruence relations for every primes dividing N. Since we only tried a finite number of discriminant (the first 18 that satisfies the SHH, the first 12 that satisfy HH but not SHH), the results discussed here are uncertain in nature, but they indicate some nice behavior.

conductor IP SIP
< 1000 387 183
1000 − 2000 31 22
2000 − 3000 7  4

An interesting thing to note is that the conductor of the integral curves of conductor bigger than 701 all have a valuation greater or equal to 3 at some prime (forcefully 2 or 3). Furthermore of the 162 conductors (totalling 356 curves) less than 701 with integral curves, 72 (totalling 232 curves) have high valuation at 2 or 3 and only 50 (totalling 64 curves) are squarefree. In fact, the last curve whose conductor is squarefree to have this property is 238B1. Lastly, even if at first it seems all the integral curves of rank 1 have Heegner points that are defined over the order, 11 of them do not have it and 6 rank 0 curve do.

Lastly, note that whenever a weak Weil curve (one that is not directly obtained by the modular parametrization) satisfies the integrality property, the strong Weil curve also does.

2.5.3 Trivial points

Even if we know that Heegner points are algebraic, one of the disappointing occurrence is when it is a point at infinity. This means that points that are not cusp in X_{0}(N) yield a trivial point (recall that cusp always give torsion points). This property of the parametrization (mapping ”non-cusp” to O) is never in conjunction with integrality. It has been observed that the Heegner points of a discriminant satisfying HH but not SHH are more frequently points at infinity.

2.5.4 The point P_{K}

If we sum up all the Heegner points for a given discriminant D, then we get a point P_{K} which is in K the quotient field of the order of discriminant D. Due to the theorem of Kolyvagin, we know that if the rank of the curve (over K) is greater than one then P_{K} is torsion. However, we observed that all the curves of rank 2 (there are no curve with bigger rank for conductors less than 3000) considered P_{K} = O, the point at infinity. It thus consolidate the fact that Heegner points (restricted to quadratic fields) are purely rank one phenomenon, being a completely trivial construction for higher ranks. Another interesting fact is that for curves of rank 1, the points P_{K} are always real. This suggests that the P_{K} have complex coordinates only when the rank of the curve (over ℚ) is zero (as in example 2.3). One could think that this is expected by Kolyvagin: when P_{K} is non-torsion then it generates a finite index group in E(K) and E(ℚ) is already such a subgroup when the curve is of rank 1 over ℚ. However if P_{K} is torsion, nothing guarantees that it should be real, and in general there is no reason to believe that the torsion-free part over ℚ is equal to the torsion-free part over K.

44  Heegner Points

Conclusion

In this thesis, we described a method to construct algebraic points on any elliptic curve, using complex multiplication and the modular parametrization. This construction can be made explicit by simple computations on binary quadratic forms and coefficients of the L-series of the elliptic curve. The results of these computations raise a number of questions.

1. Does the fact that the modular parametrization does not map non-cuspidal point of X_{0}(N) to O imply the integrality property?

2. Is there a finite number of elliptic curves (with squarefree, cube free conductor) with the integrality property? And if so, how complete is the list given?

3. Why is P_{K} real for any discriminant when we consider a rank one curve?

There are some indications that the first question can be answered positively, which might also explain why weak Weil curves rarely have this property. There is still work to do before modular parametrization is fully understood.

46  Conclusion

Appendix A

Binary Quadratic Forms

Binary quadratic forms are a concrete object which are intimately related to the ideals in quadratic orders. They are a very useful tool (if not the most convenient) to make explicit operations on those ideal. As is seen in the second chapter they are also useful to calculate Heegner Points.

Definition A.1. A binary quadratic form is a function f of two variables of the form f(X, Y) = AX^{2} + BXY + CY^{2} with A, B, C ∈ ℤ. We say it is

primitive when gcd(A, B, C) = 1
definite when Disc(f) := B^{2} − 4AC < 0 and A > 0.
reduced when − A < B ≤ A ≤ C and if A = C then B ≥ 0

(A.1)

Furthermore, we say n ∈ ℤ is represented by f if ∃X_{n}, Y_{n} ∈ ℤ such that f(X_{n}, Y_{n}) = n. If gcd(X_{n}, Y_{n}) = 1 then n is said to be properly represented.

Since we will restrict our attention to positive primitive binary quadratic form, they will henceforth only be called forms. A classical question that led to the study of these forms was to find what were the integers represented by a given form. The first thing to remark is that if we make an invertible change of variable then this set remains invariant. So we define an action of SL_{2}(ℤ) on the binary quadratic forms as follows:

( \begin{matrix}a & b \\ c & d \end{matrix}) · f(X, Y) = f(aX + bY, cX + dY) = A′X^{2} + B′XY + C′Y^{2}

where A′ = a^{2}A + acB + c^{2}C, B′ = 2(abA + bcB + cdC) + B, C′ = b^{2}A + bdB + d^{2}C

and B^{′2} − 4A′C′ = B^{2} − 4AC

(A.2)

We say that 2 forms f, g are properly equivalent if there is an M ∈ SL_{2}(ℤ) such that M · f = g. Two equivalent forms represent the same integers, however the converse is not true (take M ∈ GL_{2}(ℤ) with Det(M) = −1). The reason to favor SL_{2}(ℤ) over GL_{2}(ℤ), is that the former sends positive (resp. primitive) forms to positive (resp. primitive) forms, but not the latter.

Theorem A.1. Every form is equivalent to a unique reduced form.

Proof. Let f be a form and D = Disc(f), then it is convenient to define

τ_{f} = \frac{−B + \sqrt{D}}{2A} ∈ H = \{z ∈ ℂ | ℑ(τ) > 0\}  (A.3)

First, we can show that f ↦ τ_{f} is a injection. Since f(X, 1) is a minimal polynomial for τ over ℤ, if g is also mapped to τ_{f} then g(X, Y) = kf(X, Y) as g(X, 1) has τ as a solution and is of degree 2. If |k| > 1, then g is not primitive, and if k = −1 then g is not positive definite. Therefore g = f, and we have a bijection between quadratic points in H and forms. One can check that for M ∈ SL_{2}(ℤ),

τ_{M·f} = \frac{aτ + b}{cτ + d} =: M · τ  (A.4)

and that ℑ(M · τ) > 0. We will show that under this action any point τ in the complex upper half-plane H, there is a unique point in

F = \{τ ∈ H | −\frac{1}{2} ≤ ℜ(τ) < \frac{1}{2}, |τ| ≥ 1 and if |τ| = 1 then ℜ(τ) ≤ 0\}  (A.5)

\underline{49}

that is the image of N · τ, for some N ∈ SL_{2}(ℤ). Once this is established, it is easy to see that the form associated of such points will give reduced forms. Existence. To show this first note that for A ∈ M_{2}(ℤ) then

ℑ (\frac{aτ + b}{cτ + d}) = Det(A)|cτ + d|^{−2}ℑ(τ)  (A.6)

Since the function F (c, d) = |cτ + d|^{2} = Cc^{2}/A − Bcd/A + d^{2} admits a minimum for c, d ∈ ℤ with gcd(c, d) = 1 (else divide c and d by their gcd), we can find some a, b ∈ ℤ such that ad − bc = 1, so M = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ). Let τ′ = M_{1} · τ, and T_{n} = ( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) ∈ SL_{2}(ℤ), then T_{n}·τ′ = τ′+n and in particular ℜ(T_{n}·τ′) = Re(τ′)+n. Take n such that −\frac{1}{2} ≤ ℜ(τ′) + n < \frac{1}{2}, and write τ′′ = T_{n} · τ′ = T_{n}M · τ. Note that M_{n} = T_{n}M has the same lower row than M, thus ℑ(τ′′) is maximal amongst equivalent representatives of τ. Consequently |τ′′| ≥ 1 else ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) · τ′′ = |τ′′|^{−1}(−ℜ(τ′′) + ıℑ(τ′′)) would have greater imaginary part. If |τ′′| = 1 and \frac{1}{2} > ℜ(τ′′) ≥ 0 then ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) · τ′′ will have the same real part time −1 and thus will be in F. Otherwise τ′′ ∈ F. Unicity. Suppose that there is τ ∈ F such that ∃z = x + ıy ∈ H and M = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ) such that M · z = τ. First if c = 0 then a = d = 1 so τ = z + b. Since both have a real part smaller than 1/2 in absolute value, b = 0 and τ = z. If c ≠ 0 then we note that for any ν ∈ F, ℑ(ν) ≥ \sqrt{3}/2, and also that |cz + d| ≥ cy, so

\frac{\sqrt{3}}{2} ≤ ℑ(ν) = \frac{y}{|cz + d|^{2}} ≤ \frac{1}{c^{2}y} ≤ \frac{2}{c^{2}\sqrt{3}}  (A.7)

Consequently c^{2} ≤ 4/3 ⇒ c = ±1. Since −M and M have the same action, we can suppose c = 1. Then M = ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ( \begin{matrix}1 & d \\ 0 & 1 \end{matrix} ). So if we let z′ = z + d and ν′ = ν − a then |z′| ≥ |z| ≥ 1 and |ν′| ≥ |ν| ≥ 1. However if |z′| = 1 (resp. |ν′| = 1) then d = 0 (resp. a = 0). Furthermore, \frac{−1}{z′} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) z′ = ν′ implies that |ν′| = |z′| = 1 and so a = d = 0. Thus M = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ), and |ν| = |z| = 1. But \frac{−1}{z} = ν also implies that 0 ≥ x = −ℜ(ν) so ν ∈ F, |ν| = 1 and ℜ(ν) ≥ 0 a contradiction.

An interesting thing about the set of reduced forms is that it is finite for a fixed D. Indeed for reduced forms, D = B^{2} − 4AC ≤ −3A^{2}, so A ≤ \sqrt{−D/3}. Thus there at most −2D/3 possible choices of A and B (C being determined by (B^{2} − D)/4A).

Remark A.1. Also for any integer n we can find a form f in any equivalence set such that gcd(A, n) = 1. Indeed, let A′ bet the X^{2} coefficient in ( \begin{matrix}a & b \\ c & d \end{matrix}) · f(X, Y), then A′ = a^{2}A + acB + c^{2}C. So we take a, c such that for every prime p | n,

if p ∤ A  then p ∤ a and p | c
if p | A and p ∤ C then p | a and p ∤ c
if p | A and p | C then  p ∤ ac

(A.8)

so that p ∤ A′ and gcd(a, c) = 1. b and d are the chosen so that ad − bc = 1.

There is another operation that one can do on forms, namely composition. Though Gauss defined it in a very general way, it is also very clumsy to work with. Later, Dirichlet gave an equivalent description of it which is more simple but does not apply for arbitrary forms. Definition A.2. Let f(X, Y) = AX^{2} +BXY +CY^{2} and g(X, Y) = A′X^{2} +B′XY + C′Y^{2} be two forms of discriminant D, and such that gcd(A, A′, \frac{B+B′}{2} ) = 1. The Dirichlet composition of f and g is defined as the form h(X, Y) = AA′X^{2} + B′′XY + \frac{B^{′′2}−D}{4AA′}Y^{2}, with B′′ such that

B′′ ≡ B (mod 2A)
B′′ ≡ B′ (mod 2A′)
B^{′′2} ≡ D (mod 4AA′)

(A.9)

Though this definition is not general enough to encompass all the pairs of forms of a given discriminant, we can use it to define the composition of reduced forms. First, given two reduced forms, it is possible to make their X^{2} coefficients coprime by changing one of them to another equivalent form. Then the composition is possible using the above definition. Finally, it suffice to find the reduced form equivalent to this composition.

\underline{51}

Theorem A.2. Let the form class group be

Cl(D) = {equivalence classes of forms of discriminant D}  (A.10)

Then Dirichlet composition makes Cl(D) into an abelian group.

Though we will not show the proof of this theorem ([4] p.51), one can easily see that the identity is X^{2} + αXY + \frac{α−D}{4}Y^{2} (where D ≡ α (mod 4)). Furthermore, the inverse of AX^{2} + BXY + CY^{2} is AX^{2} − BXY + CY^{2}, as can be seen by acting with ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) on the latter.

The association f ↦ τ_{f} almost bridges forms of discriminant D and O ideals for Disc(O) = D. Indeed, one can define a map I from forms to lattices as follows I(f) = A⟨1, τ_{f}⟩, where A is the X^{2} coefficient in f. In fact we already saw in lemma 1.2 that O = ⟨1, Aτ_{f}⟩, thus I(f) is actually an integral O ideal and its norm is easily seen to be A. Again in lemma 1.2, we saw that this ideal has an inverse, the fractional O ideal ⟨1, \overline{τ_{f}}⟩ where \overline{τ_{f}} is the complex conjugate of τ_{f}.

Lemma A.1. The map I : Cl(D) → Cl(O) is a bijection. In particular Cl(O) is finite.

Proof. Injectivity. Let K = Quot(O), we will show that

\begin{matrix}τ′ = \frac{aτ+b}{cτ+d} & for ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ) \\ ⇔ & ⟨1, τ⟩ = λ⟨1, τ′⟩ for λ ∈ K^{×} \end{matrix}  (A.11)

If τ′ = ( \begin{matrix}a & b \\ c & d \end{matrix}) · τ then

⟨1, τ′⟩ = (cτ + d)⟨cτ + d, aτ + b⟩ = cτ + d⟨1, τ⟩  (A.12)

since ( \begin{matrix}a & b \\ c & d \end{matrix}) defines an invertible change of variable. If ⟨1, τ⟩ = λ⟨1, τ′⟩, then

\begin{aligned}λτ′ & = aτ + b \\ λ & = cτ + d \end{aligned} where ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(ℤ)  (A.13)

However by equation A.2, if Det ( \begin{matrix}a & b \\ c & d \end{matrix}) < 0 then τ′ ∉ H a contradiction so ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ). Since f and g are equivalent if and only if τ_{f} is equivalent to τ_{g}, it follows that I(f) is homothetic to I(g) if and only if f and g are equivalent, proving injectivity. Surjectivity. Write the fractional ideal 𝔞 as ⟨α, β⟩, then (up to switching α and β) one has that 𝔞 ≃ ⟨1, τ⟩ for τ = α/β ∈ H. τ admits a minimal polynomial of degree 2 over ℤ, say AX^{2} + BX + C with gcd(A, B, C) = 1. Let f(X, Y) = AX^{2} + BXY + CY^{2}, then τ = τ_{f} and ⟨1, τ⟩ = A^{−1}τ_{f}

This bijection is extremely useful, as it enables us to extend results we have on forms to ideal, for example:

Corollary A.1. Let O be an order in a quadratic number field. Then Cl(O) is finite.

Corollary A.2. Let 𝔞 be an integral O ideal, and k ∈ ℤ then there is a equivalent integral ideal 𝔟 such that no prime dividing k divides 𝔟 (𝔟 is coprime to k).

The map I : Cl(D) → Cl(O) actually gives an isomorphism of group, but is rather long to prove (see [4] §.7).

Remark A.2. One can very efficiently obtain the form class group simply by enumerating all the possible values of A (recall 0 < A ≤ \sqrt{−D/3}) and B (−A < B ≤ A) for which (B^{2} − D)/4A is an integer. For example, take D = −31 (\sqrt{−D/3} = 3.21 …). One easily find that the pairs (A, B) satisfying these conditions are (1, 1), (2, −1) and (2, 1).

For completeness, one can check that an inverse map I^{−1} can be defined as follows: given 𝔞 = ⟨α, β⟩ then

I^{−1}(𝔞) = \frac{N_{K/ℚ}(αX + βY)}{N_{K/ℚ}(𝔞)}  (A.14)

\underline{53}

But note that I ∘ I^{−1} is not the identity map on ideals, but on equivalence class of ideals, I^{−1} being defined up to a change of basis for 𝔞.

As a conclusion to this appendix, we will show two lemmas that will be useful to determine conditions for the existence of Heegner points. Lemma A.2. Let f(X, Y) = AX^{2} + BXY + CY^{2} be a quadratic form, then f represents N ∈ ℤ properly if and only if f is equivalent to NX^{2} + B′XY + C′Y^{2}. Proof. If f is equivalent to NX^{2} + B′XY + C′Y^{2}, then for some ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ), ( \begin{matrix}a & b \\ c & d \end{matrix}) · f(1, 0) = f(a, c) = N and gcd(a, c) = 1. So N is properly represented.

If N is properly represented then there exists a, c ∈ ℤ such that f(a, c) = N and gcd(a, c) = 1. Take b, d ∈ ℤ such that ad − bc = 1, then ( \begin{matrix}a & b \\ c & d \end{matrix}) · f(X, Y) = NX^{2} + B′XY + C′Y^{2} by equation A.2. Lemma A.3. The following are equivalent: 1− There is 𝔫 ◁ O such that O/𝔫 = ℤ/N. 2− There is a primitive binary quadratic form with discriminant equal to D = Disc(O) which properly represents N. 3− The equation D = B^{2} − 4NC has a solution with gcd(N, B, C) = 1 Proof. (3 ⇒ 2) If the solution to the equation exist the f(X, Y) = NX^{2}+BXY +CY^{2} is a form of discriminant D properly representing N. (2 ⇒ 3) If such a form exists find its equivalent form whose X^{2} coefficient is N. Its discriminant is of the form B^{2} − 4NC with gcd(N, B, C) = 1. (2 ⇒ 1) We saw that O/I(f) is ℤ/f(1, 0), so it is sufficient to take the ideal associated to the form NX^{2} + BXY + CY^{2} ∈ [f]. (1 ⇒ 2) Let 𝔫 be such that O/𝔫 = ℤ/N, and let A^{2} + BXY + CY^{2} = f(X, Y) = I^{−1}(𝔫). Then for some α ∈ K^{×}, 𝔫 = α⟨1, τ_{f}⟩. Now

\begin{aligned}A𝔫 = αA⟨1, τ_{f}⟩ & ⇒ & A^{2}N_{K/ℚ}(𝔫) = N_{K/ℚ}(α)A \\ & ⇒ & N_{K/ℚ}(𝔫) = N_{K/ℚ}(α)/A \end{aligned}  (A.15)

However, α⟨1, τ_{f}⟩ = 𝔫 ⊆ O = ⟨1, Aτ_{f}⟩, consequently,

\begin{aligned}ατ_{f} & = a + bAτ_{f} \\ α & = c + dAτ_{f} \end{aligned} ⇒ (c + dAτ_{f})τ_{f} = a + bAτ_{f}  (A.16)

Using Aτ_{f}^{2} = −Bτ_{f} − C, we get that c = Ab + Bd and Cd + a = 0. On the other hand

A^{−1}N_{K/ℚ}(α) = A^{−1}(c^{2} − Bcd + ACd^{2}) = ⋯ = Ab^{2} + Bab + Ca^{2} = f(b, a) (A.17)

If f(b, a) = k > 1, then k | c and k | d so

𝔫 = α⟨1, τ⟩ = ⟨c + dAτ_{f}, a + bAτ_{f}⟩ = k𝔞  (A.18)

and O/𝔫 = O/k𝔞 contains a subgroup of the form (ℤ/kℤ)^{2}, a contradiction. So gcd(a, b) = 1 and f properly represents N.

Appendix B

List of integral curves

Here are the names, referring to Cremona’s table (see [5] or his website), of the elliptic curves that appear to have the integrality property. As mentioned before (see 2.5.2) the integrality of the Heegner points has been verified for the first 18 discriminants satisfying SHH and the first 12 discriminants satisfying HH but not SHH.

11A1, 14A1, 14A2, 15A1, 15A4, 17A1, 19A1, 20A1, 20A3, 21A1, 21A3, 24A1, 24A2, 24A3, 24A5, 24A6, 26A1, 26B1, 27A1, 30A1, 30A2, 32A1, 34A1, 35A1, 36A1, 36A2, 36A3, 36A4, 37A1, 38B1, 39A1, 39A2, 40A1, 40A2, 40A4, 42A1, 43A1, 44A1, 44A2, 45A1, 45A2, 48A1, 48A2, 48A3, 48A5, 48A6, 49A1, 49A2, 50A1, 50B1, 50B2, 51A1, 52A1, 53A1, 54A1, 54B1, 54B3, 55A1, 55A2, 56A1, 56A2, 56A3, 56B1, 56B2, 57A1, 58A1, 61A1, 62A1, 64A1, 64A2, 64A3, 65A1, 65A2, 66A1, 66B1, 66B2, 69A1, 69A2, 70A1, 70A2, 72A1, 72A2, 75C1, 76A1, 77A1, 79A1, 80A1, 80A3, 80A4, 80B1, 82A1, 83A1, 84A1, 84A2, 84B1, 84B2, 88A1, 89A1, 90A1, 90B1, 91A1, 92A1, 92A2, 92B1, 94A1, 94A2, 96A1, 96A2, 96A3, 96A4, 96B1, 96B2, 96B3, 96B4, 99A1, 99A2, 101A1, 102A1, 104A1, 105A1, 105A2, 108A1, 108A2, 110B1, 112A1, 112A2, 112B1, 112B2, 112B4, 112C1, 118A1, 120A1, 120A2, 120B1, 120B2, 123B1, 124A1, 124B1, 126A1, 128A1, 128A2, 128B1, 128C1, 128C2, 128D1, 130B1, 131A1, 132A1, 132A2, 135A1, 136A1, 136B1, 138A1, 140A1, 141D1, 142B1, 143A1, 144A1, 144A2, 144A3, 144A4, 144B1, 144B2, 145A1, 150A1, 152A1, 152B1, 153A1, 155C1, 156A1, 156B1, 160A1, 160A2, 160B1, 160B2, 162A1, 162B1, 162C1, 162D1, 168A1, 168A2, 176A1, 176B1, 176C1, 180A1, 180A2, 184A1, 184B1, 189A1, 190B1, 192A1, 192A2, 192A4, 192B1, 192B2, 192B4, 192C1, 192C2, 192D1, 192D2, 196A1, 200B1, 200E1, 204B1, 208B1, 208C1, 210D1, 216A1, 216B1, 220B1, 224A1, 224A2, 224B1, 224B2, 225A1, 225A2, 234C1, 236A1, 238B1, 240A1, 240A2, 240C1, 240C2, 240D1, 243A1, 243A2, 243B1, 248A1, 248C1, 256A1, 256A2, 256B1, 256B2, 256C1, 256C2, 256D1, 256D2, 264A1, 264B1, 272A1, 272C1, 280A1, 288A1, 288A2, 288D1, 288D4, 297B1, 300A1, 304C1, 304D1, 304F1, 312A1, 312B1, 312C1, 320A1, 320B1, 320C1, 320D1, 320D2, 320E1, 320E2, 320F1, 324B1, 324C1, 325B1, 336A1, 336A2, 336B1, 336B2, 336F1, 336F2, 342E1, 348A1, 348B1, 350C1, 360B1, 368C1, 368D1, 368E1, 368E2, 368F1, 378D1, 384A1, 384A2, 384B1, 384B2, 384C1, 384C2, 384D1, 384D2, 400D1, 400H1, 405C1, 405F1, 416A1, 416B1, 420D1, 425C1, 432A1, 432A3, 432B1, 432B2, 432D1, 432H1, 440B1, 459A1, 480A1, 480B1, 480C1, 480G1, 486A1, 486B1, 496A1, 496B1, 496D1, 496E1, 504A1, 504E1, 528A1, 528D1, 528E1, 528E2, 540B1, 540F1, 544A1, 544D1, 558A1, 560A1, 560F1, 567A1, 575A1, 576A1, 576A3, 576D1, 576E1, 576E3, 576F1, 576F2, 576H1, 576H2, 576I1, 600B1, 624A1, 624C1, 624D1, 624G1, 624J1, 640C1, 640G1, 648A1, 648B1, 672A1, 672G1, 675B1, 700B1, 700C1, 702B1, 704A1, 704B1, 704C1, 704G1, 704J1, 704K1, 720A1, 720I1, 768B1, 768D1, 768E1, 768F1, 768G1, 768H1, 784I1, 800B1, 800F1, 816G1, 864A1, 864D1, 880A1, 880H1, 896B1, 896C1, 944F1, 960A1, 960F1, 972A1, 972B1, 972C1, 1008B1, 1008D1, 1080A1, 1080K1, 1152B1, 1152D1, 1200F1, 1200O1, 1216A1, 1216E1, 1296B1, 1296C1, 1296G1, 1296K1, 1296L1, 1344C1, 1344F1, 1350A1, 1392L1, 1392O1, 1440B1, 1440I1, 1680O1, 1728A1, 1728A2, 1728C1, 1728H1, 1728R1, 1728V1, 1728V3, 1728AA1, 2160C1, 2160G1, 2160Q1, 2160X1, 2304B1, 2304I1, 2700L1, 2800P1, 2800W1

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