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An Elementary and Computational Approach to Heegner Points

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An Elementary and Computational Approach to Heegner Points AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS SHELLY MANBER Abstract. In this paper we present a method for explicitly computing ratio- nal points on elliptic curves using Heegner points. This method was crucial to the proof of Gross-Zagier, which proves the rank one case of the Birch and Swinnerton-Dyer Conjecture. Although this use of Heegner points can be found in many books and articles, we strive here to present it in a more concrete and complete form, and using explicit and elementary tools. 1. Introduction One of the most basic and fundamental mathematical problems is to find an integral solution to a given polynomial equation. This is a problem most people encounter even in grade school. For example, it is a classic and relatively simple exercise to enumerate all pythagorean triples, or integral solutions to the equation x2 +y2 = z2. By contrast, Fermat's Last Theorem - prove that there are no integral solutions to the equation xn + yn = zn for n > 2 - is a famously difficult problem that took hundreds of years to solve. Indeed, although it is simple to state and understand, finding integral solutions to arbitrary polynomial equations remains an extremely difficult problem in modern number theory. We will be particularly interested in finding rational points on elliptic curves, or smooth, one dimensional, genus one projective varieties. Elliptic curves are a natural choice of curve for study because they are simple enough that much is known about them and they can be worked with explicitly, yet complex enough that there is a rich theory attached to them with connections to many different fields. For example, elliptic curves are abelian algebraic groups and can be described analytically as a quotient of C such that the group law corresponds to addition in C and algebraic morphisms correspond to homotheties on C. Furthermore, to each elliptic curve we can assign an L-series and modular form that play a role in analytic number theory. We outline here a method to construct explicit rational points on elliptic curves defined over Q. The method is a consequence of the modularity theorem (for- merly known as the Taniyama-Shimura-Weil conjecture) and the Artin Reciprocity Theorem. In Section 2, we will present background information on elliptic curves and their moduli space from an analytic perspective. In Section 3, we will present background information on elliptic curves and their moduli space viewed as algebraic varieties. In Section 4 we will discuss the theory of complex multiplication and in Section 5 some background on orders in imaginary quadratic number fields, both of which we will need to discuss the Artin Reciprocity Theorem in Section 6. In Section 7 we will define Heegner points and generalize the Artin Reciprocity Theorem to 1 2 SHELLY MANBER construct explicit finite sets of points on modular curves that are defined over a finite algebraic extension of Q and form a complete set of galois conjugates. This, together with the modularity theorem, which we will state in Section 8 and which gives us a morphism to each elliptic curve defined over Q from a modular curve determined by its conductor, will complete our algorithm for constructing rational points on elliptic curves. In Section 9 we will present some examples of this algorithm computed with PARI, a computer algebra system specializing in number theory computations. 2. Elliptic Curves and Their Moduli Space - Anayltically In this section we present elementary background information about elliptic curves and the modular curves X0 and X0(N). A positively oriented lattice Λ in C is an additive subgroup of C of the form Λ = fm!1 + n!2jm; n 2 Zg where !1 and !2 are fixed complex numbers such that !1=!2 2 H, the upper half plane. If Λ = fm!1 + n!2jm; n 2 Zg then we say that the pair (!1;!2) is a basis for Λ and denote this by Λ = h!1;!2i. It follows directly from the definitions that two pairs of complex numbers (!1;!2) and (τ1; τ2) are bases for the same lattice if and only if !1 a b τ1 a b = for some 2 SL2(Z): !2 c d τ2 c d We say that two lattices Λ and Λ0 are homothetic if Λ = αΛ0 for some α 2 C. Thus h!1;!2i and hτ1; τ2i are homothetic if and only if there exists an α 2 C such that h!1;!2i = hατ1; ατ2i. We can define an equivalence relation on the set of positively oriented lattices by assigning two lattices to be equivalent if they are homothetic; let L denote the set of equivalence classes of lattices under this equivalence relation. Since h!1;!2i s !2h!1=w2; 1i, each element of L can be represented by a lattice with a basis of the form hτ; 1i with τ 2 H. Furthermore, τ is unique up to fractional linear transformation by SL2(Z). Indeed, every basis for the lattice h!1;!2i is of a b the form ha! + b! ; c! + d! i for some 2 SL ( ), and thus the set of 1 2 1 2 c d 2 Z lattices of the form hτ; 1i equivalent to h!1;!2i is exactly a!1 + b!2 a b ; 1 2 SL2(Z) : c!1 + d!2 c d Thus the elements of L are in bijective correspondence with the points of H=SL2(Z), where SL2(Z) acts on H by fractional linear transformation. An elliptic curve is an analytic manifold given by C=Λ where Λ is a lattice in C. It naturally has the structure of an analytic manifold with charts given by lifting an open subset of C=Λ to any connected open subset of C in the preimage of the projection from C to C=Λ. It also has a natural group structure given by addition in C. Claim 2.1. Two elliptic curves are isomorphic as analytic manifolds if and only if their corresponding lattices are homothetic. Proof. See [Sil86, Chapter VI, Corollary 4.4.1]. AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 3 From Claim 2.1 we see that the set of elliptic curves up to isomorphism is in bijective correspondence with the elements of L and hence with the set H=SL2(Z). Let X0 denote the analytic manifold associated to H=SL2(Z) with charts given by lifting to the upper half plane, as with C=Λ. Then we say that the points of X0 index the set of elliptic curves up to isomorphism, i.e. each point corresponds to a unique isomorphism class of elliptic curves and each isomorphism class of elliptic curves corresponds to a point on X0. We will see in Section 3 that both E and X0 have natural structures as algebraic varieties over Q. The curve X0 is a special case of a family of algebraic curves called modular curves, denoted X0(N) with N 2 N. They are of particular interest here because of the Modularity Theorem, which states that for every elliptic curve E defined over the rationals, there is a positive integer N (called the conductor of the curve; see Section 8) and a morphism of algebraic varieties ΦE : X0(N) ! E defined over Q. Thus we can construct explicit points on E over a number field K by constructing K-rational points on X0(N) and taking their image under ΦE. To define the curves X0(N), we first need to define the modular congruence subgroups Γ0(N). Definition 1. Let N be a positive integer. Then n a b o Γ (N) := 2 SL ( ) j c ≡ 0 mod N 0 c d 2 Z and X0(N) := H=Γ0(N), where Γ0(N) acts on H by fractional linear transformation. The curve X0(N) is called the modular curve of level N and has a structure of an analytic manifold, with charts given as with X0. Note that Γ0(N) is a subgroup of SL2(Z) and that Γ0(1) = SL2(Z) so that X0(1) = X0. Just as the points X0 index elliptic curves in C up to homothety, the points of 0 X0(N) index pairs of elliptic curves (E; E ) together with a cyclic isogeny of level N 0 between them [Gro84]. In other words, the points of X0(N) index triples (E; E ;'), up to equivalence, where ' is a surjective analytic map ' : E ! E0 such that ker 0 (') is an additive cyclic subgroup of E of order N. The triples (E1;E1;'1) and 0 (E2;E2;'2) are defined to be equivalent if there are isomorphisms γ : E1 ! E2 and 0 0 0 γ : E1 ! E2, such that the following diagram commutes: '1 0 (1) E1 / E1 : γ γ0 '2 0 E2 / E2 Equivalently, we can say that each point of X0(N) corresponds to an elliptic curve with a distinguished cyclic subgroup of order N, up to isomorphisms that preserve the subgroup. These notions are equivalent because clearly the map ' : E ! E0 uniquely determines the subgroup ker(') of E, and additionally the pair (E, ker(')) uniquely determines E0 up to isomorphism [Sil86, Chapter III, Prop 4.12]. To prove the bijective correspondence between points of X0(N) and elliptic curves with distinguished subgroups, we make two claims (adopted from [DS05, Theorem 1.5.1(a)]). 4 SHELLY MANBER 1 Claim 2.2. For any τ 2 H, let [Eτ ; N ] denote the equivalence class of the elliptic 1 curve C=hτ; 1i and subgroup generated by N + hτ; 1i (as above, two curve-subgroup pairs are equivalent if there is an isomorphism of elliptic curves that preserves the subgroup).
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