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 rational 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 = {mω1 + nω2|m, n ∈ Z} where ω1 and ω2 are ﬁxed complex numbers such that ω1/ω2 ∈ H, the upper half plane. If Λ = {mω1 + nω2|m, n ∈ Z} then we say that the pair (ω1, ω2) is a basis for Λ and denote this by Λ = hω1, ω2i. It follows directly from the deﬁnitions that two pairs of complex numbers (ω1, ω2) and (τ1, τ2) are bases for the same lattice if and only if ω1 a b τ1 a b = for some ∈ SL2(Z). ω2 c d τ2 c d

We say that two lattices Λ and Λ0 are homothetic if Λ = αΛ0 for some α ∈ C. Thus hω1, ω2i and hτ1, τ2i are homothetic if and only if there exists an α ∈ C such that hω1, ω2i = 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 hω1, ω2i ∼ ω2hω1/w2, 1i, each element of L can be represented by a lattice with a basis of the form hτ, 1i with τ ∈ H. Furthermore, τ is unique up to fractional linear transformation by SL2(Z). Indeed, every basis for the lattice hω1, ω2i is of a b the form haω + bω , cω + dω i for some ∈ SL ( ), and thus the set of 1 2 1 2 c d 2 Z lattices of the form hτ, 1i equivalent to hω1, ω2i is exactly aω1 + bω2 a b , 1 ∈ SL2(Z) . cω1 + dω2 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 ∈ 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) := ∈ SL ( ) | 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 , ϕ), up to equivalence, where ϕ is a surjective analytic map ϕ : E → E0 such that ker 0 (ϕ) is an additive cyclic subgroup of E of order N. The triples (E1,E1, ϕ1) and 0 (E2,E2, ϕ2) 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:

ϕ1 0 (1) E1 / E1 .

γ γ0

ϕ2 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 ϕ : E → E0 uniquely determines the subgroup ker(ϕ) of E, and additionally the pair (E, ker(ϕ)) 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 ∈ 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). Then the set of all equivalence classes of elliptic curves paired with order 1 N cyclic subgroups is exactly {[Eτ , N ]: τ ∈ H}.

1 1 Claim 2.3. Let τ, ω ∈ H. Then [Eω, N ] = [Eτ , N ] if and only if there exists a b ∈ Γ (N) such that ω = aτ+b . c d 0 cτ+d

1 From Claim 2.3 we see that the distinct elements of {[Eτ , N ]: τ ∈ H} are in 1 bijection with H/Γ0(N), and from Claim 2.2 that the set {[Eτ , N ]: τ ∈ H} is in bijection with the the set of all elliptic curves with distinguished cyclic, order N subgroups. Thus the correspondence follows immediately from the two claims. We now set about proving the claims. Proof of Claim 2.2 . Let E = C/Λ be an elliptic curve with cyclic subgroup G of order N, and let x + Λ be any generator of G. Since G has order N, we have Nx + Λ = Λ so Nx ∈ Λ. Our goal will be to find a specific generator x + Λ of G such that we can complete Nx to a basis for Λ, i.e. write Λ = hτ, Nxi. If we 1 can successfully do this then we apply the homothety of multiplication by Nx to τ ∼ Λ to get an isomorphic curve C/h Nx , 1i = E. This homothety sends x + Λ to 1 τ τ 1 1 N + h Nx , 1i. Hence the pair (E,G) is equivalent to (C/h Nx , 1i, h N i) = [Eτ 0 , N ], 0 τ where τ = Nx . All that remains is to find complex numbers x and τ such that x + Λ generates G and Λ = hτ, Nxi. Let x0 be a representative of any generator of G. Then (C/Λ)/G = C/L where L = {ω + kx0|ω ∈ Λ, k ∈ Z}. The set L is a discrete subgroup of C and contains Λ, so it is a lattice in C. If there exists y ∈ L such that x0 = ry for some integer r, then y + Λ is also a generator of G. Indeed, y has order dividing N because y = ωy + kyx0 with ωy ∈ Λ, ky ∈ Z, and Nx0 ∈ Λ, so Ny ∈ Λ. If y had order s < N then sy ∈ Λ, so we would have sx0 = rsy ∈ Λ, which would be a contradiction to x0 representing a generator. This means that we can always choose a complex number x such that x + Λ generates G and x is indivisible in L, i.e. x is not an integral multiple of any element of L. We next claim that choosing such an x guarantees that Nx is indivisible in Λ. Indeed, assume to the contrary that there exists y ∈ Λ such that ry = Nx. Then N we must have gcd(N, r) = 1, since otherwise we would have gcd(N,r) x ∈ Λ and N gcd(N,r) ∈ Z, which is impossible since x generates a cyclic group of order N. Let Na Nb L = hω1, ω2i and let x = aω1 + bω2 with a, b ∈ Z, so y = r ω1 + r ω2 ∈ Λ ⊂ L. Since gcd(N, r) = 1 we must have r dividing both a and b. But a and b cannot share any common factors since x in indivisible in L. So r = 1 and Nx indivisible in Λ. Finally, let hλ1, λ2i be a basis for Λ and Nx = aλ1 +bλ2. Since Nx is indivisible, r s we have gcd(a, b) = 1 so there exist r, s ∈ such that rb−sa = 1. Now ∈ Z a b SL2(Z) so hrλ1 + sλ2, aλ1 + bλ2i is an alternate basis for Λ. Letting τ = rλ1 + sλ2, we have Λ = hτ, Nxi as desired. AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 5

1 1 Proof of Claim 2.3 . Let [Eω, N ] = [Eτ , N ]. Then there exists an isomorphism 1 ϕ : Eτ → Eω such that the group generated by N on Eτ is sent to the group 1 generated by N on Eω. Note that if Eτ or Eω have automorphisms then there will be more than one isomorphism between the two curves, but at least one will 1 1 send the group generated by N on Eτ to the group generated by N on Eω. Let a b ∈ SL ( ) be the matrix associated with ϕ, i.e, ω = aτ+b and the c d 2 Z cτ+d 1 isomorphism ϕ : C/hτ, 1i → C/hω, 1i is given by multiplication by cτ+d . Now since ∼ 2 1 C = R , we can write N uniquely in the form 1 (2) = r(aτ + b) + s(cτ + d) N with r, s ∈ R, so 1 (3) ϕ( ) = rω + s. N 1 By our choice of ϕ, the image of N must be a generator of the group generated by 1 1 m N on Eω. This means that ϕ( N ) ≡ N mod hω, 1i for some m relatively prime to N. Combining with (3) yields m rω + s ≡ mod hω, 1i, N m so we must have r ∈ Z and s = N + k with k ∈ Z. Replacing this expression for s in (2) gives 1 m = r(aτ + b) + ( + k)(cτ + d), N N mc from which we see that ra + N + kc = 0. Since gcd(m, N) = 1, we must have N|c. a b So ∈ Γ (N). c d 0

a b Conversely, let ω = aτ+b for some ∈ Γ (N). Since ad − bc = 1, we cτ+d c d 0 have 1 −c a = (aτ + b) + (cτ + d), N N N so 1 −c a ϕ( ) = ω + . N N N 1 a But we know that N divides c, so in fact ϕ( N ) = N in C/hω, 1i. We also know that gcd(a, c) = 1 since ad − bc = 1, which means that in particular, gcd(a, N) = 1. So 1 1 the image of N under ϕ generates the same subgroup of C/hω, 1i that N generates, 1 1 and hence [Eω, N ] = [Eτ , N ].

Notice that the proofs of Claims 2.2 and 2.3 give us an explicit correspondence between elements of H/Γ0(N) and pairs of elliptic curves with a cyclic isogeny 1 between them. Given τ ∈ H/Γ0(N), the natural projection C/hτ, 1i → C/hτ, N i is a cyclic isogeny of order N whose kernel is the subgroup of Eτ = C/hτ, 1i generated 1 1 0 by N + hτ, 1i and thus corresponds to [Eτ , N ]. Conversely, given that ϕ : E → E is an isogeny whose kernel is a cyclic order N subgroup of E, Claim 2.2 shows that 6 SHELLY MANBER

1 there exists a τ ∈ H such that [E, ker(ϕ)] is equivalent to [Eτ , N ] and Claim 2.3 tells us that such a τ is unique up to the action of Γ0(N). By symmetry, it is equivalent to assign to each τ ∈ H/Γ0(N) the pair of lattices τ 1 hτ, 1i ⊂ h N , 1i. Indeed, multiplying both lattices hτ, 1i and hτ, N i by −1/τ and 00 −1 applying the change of variables τ = τ gives a homothetic pair of lattices τ 00 h−1, τ 00i ⊂ h−1, i, N or, choosing a different basis, τ 00 hτ 00, 1i ⊂ h , 1i N 00 with τ ≡ τ mod SL2(Z). The last thing we will need from this section is the notion of lattice index. 0 0 0 Let Λ = hλ1, λ2i and Λ = hλ1, λ2i be lattices in C. There is a unique linear 2 ∼ 0 0 automorphism T of the vector space R = C sending λ1 to λ1 and λ2 to λ2. We define the lattice index of Λ in Λ0, denoted [Λ0 : Λ], to be the determinant of T [Ste04, Definition 10.3.1]. Note that if Λ ⊂ Λ0 then [Λ0 : Λ] is the order of the additive group Λ0/Λ. Lemma 2.4. If N ⊂ M ⊂ L are lattices in C then [L : N] = [L : M][M : N].

Proof. Let T1 : L → M and T2 : M → N be linear automorphisms of C with determinants d1 and d2 respectively. Then T2T1 : L → N is a linear automorphism with determinant d1d2. Thus [L : N] = d1d2 = [L : M][M : N]. Note that we have defined the notion of lattice index on two concrete lattices, not on elements of L. Thusfar, our definition of lattice index does not extend to elements of L, because applying a homothety to one lattice from a pair latices will change the determinant of the automorphism between the two lattices if the complex number associated to the homothety has a magnitude other than 1. Claim 2.5. Let E and E0 be a pair of elliptic curves with an isogeny ϕ : E → E0 such that ker ϕ is cyclic order N. Then there exist lattices Λ ⊂ Λ0 in C such that ϕ can be realized as the analytic map φ : C/Λ → C/Λ0 induced by the identity on C. Furthermore, for any such pair of lattices we have [Λ0 : Λ] = N. Proof. Let Λ be any lattice such that E = C/Λ, then we can find a lattice Λ0 such that E0 = C/Λ0 and Λ ⊂ Λ0. Indeed, we will see later on (Theorem 3.1) that every isogeny between elliptic curves C/Λ → C/Λ0 is of the form z 7→ αz for some α such 0 that αΛ ⊂ Λ0. By choosing Λ = Λ0/α as the representative for the homothety 0 class of Λ0, we have our desired Λ ⊃ Λ. Now consider the map φ : C/Λ → C/Λ0 induced by the identity on C. Since Λ ⊂ Λ0, we have ker ϕ = Λ0/Λ. As in the proof of Claim 2.2, we can choose a generator x + Λ of ker ϕ such that Λ = hτ, Nxi and Λ0 = hτ, xi for some τ ∈ C. Letting (τ, x) be a basis for C as an R-vector space, the linear automorphism θ 1 0 given by satisfies θ(Λ0) = Λ and has determinant N. 0 N Note that it is not the case that [Λ0 : Λ] = N whenever E = C/Λ and E0 = C/Λ0. Claim 2.5 shows that whenever E = C/Λ and E0 = C/Λ0 and Λ ⊂ Λ0, we must have [Λ0 : Λ] = N. AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 7

3. Elliptic Curves and Their Moduli Space - Algebraically An elliptic curve C/Λ also has the structure of a projective variety. To define the structure explicitly given the analytic formulation, we will introduce several well- known analytic functions. The following several paragraphs give a brief summary of these analytic functions and their properties. A more detailed presentation can be found in almost any textbook on analytic number theory, complex analysis, or elliptic curves, for example [SS03]. The Weierstrass elliptic function ℘ : C × L → C is defined as follows: X 1 1 ℘(z, Λ) = z−2 + − . (z − τ)2 (τ)2 τ∈Λ τ6=(0,0) One can check that ℘(z +τ, Λ) = ℘(z, Λ) for all τ ∈ Λ, so for a fixed lattice Λ the Weierstrass function is doubly periodic as a function of z. The Weierstrass elliptic function also satisfies the differential equation 0 2 (4) ℘ (z, Λ) = 4℘(z, Λ) − g2(Λ)℘(z, Λ) − g3(Λ) where the analytic functions g2 : L → C and g3 : L → C are defined by X 1 g (Λ) = 60 2 τ 4 τ∈Λ τ6=(0,0) X 1 g (Λ) = 140 . 3 τ 6 τ∈Λ τ6=(0,0) The explicit correspondence between C/Λ and it’s corresponding projective variety is given by the following map:

ϕ : C → P2(C) τ 7→ [℘(τ, Λ); ℘0(τ, Λ); 1].

The map ϕ is defined on C/Λ since ℘(z + τ, Λ) = ℘(z, Λ) for all τ ∈ Λ, and (4) shows that the image of φ is in the locus of the homogenous equation y2z = 3 2 3 4x − g2(Λ)xz − g2(Λ)z in P2(C). Notice that ℘(z, Λ) has a pole when z ∈ Λ, so the image of 0 in C/Λ is [0, 1, 0], and we call this the “point at infinity”, or “distinguished point”. It turns out that ϕ defines an isomorphism between the the 2 3 2 3 locus of y z = 4x − g2(Λ)xz − g2(Λ)z and C/Λ as analytic manifolds [Sil86, Chapter VI, Proposition 3.6(b)]. 2 3 2 3 The equation y z = 4x − g2(Λ)xz − g2(Λ)z describes a smooth, genus one projective curve over C. Furthermore, every smooth, genus one, projective curve can be expressed over C as the locus of an equation of the form y2z = 4x3−axz2−bz3 [Sil86, Chapter VI, Proposition 3.6 and Theorem 5.1]. Two other analytic functions will be important to the study of elliptic curves. These are the modular discriminant 3 2 ∆ := g2 − 27g3 and Klein’s j-invariant g3 j := 1728 2 . ∆ 8 SHELLY MANBER

0 0 4 0 Notice that if Λ = αΛ for some α ∈ C then g2(Λ ) = g2(Λ)/α and g3(Λ ) = 6 0 12 0 g3(Λ)/α . This means that ∆(Λ ) = ∆(Λ)/α and j(Λ ) = j(Λ). So homothetic lattices have the same j-invariant. Let E be an elliptic curve viewed as a projective variety over C. For each lattice ∼ 2 3 2 3 Λ such that C/Λ = E, the equation y z = 4x − g2(Λ)xz − g2(Λ)z gives an expression deﬁning E in P2(C). Since all such lattices are homothetic, there is a unique value j(Λ) associated to E. We denote this j(E) and call it the j-invariant of E. If τ ∈ C we deﬁne j(τ) to be j(hτ, 1i). We have j(τ) = j(τ 0) whenever τ ≡ τ 0 mod SL2(Z).

Now that we have an analytic and algebraic structure of elliptic curves, we move on to understanding the maps between elliptic curves. An isogeny of elliptic curves is a morphism between elliptic curves as projective varieties that sends the distinguished point of one curve to the distinguished point of the other. Every isogeny is in particular an analytic map between elliptic curves as analytic manifolds. Theorem 3.1. Let E,E0 be elliptic curves and Λ, Λ0 their associated lattices. (1) Assigning to each isogeny E → E0 the induced analytic map C/Λ → C/Λ0 gives a bijection of sets 0 0 {isogenies ϕ : E → E } ↔ {nontrivial holomorphic maps φ : C/Λ → C/Λ with φ(0) = 0}. (2) Sending α ∈ C to the map φα : C → C z 7→ αz, gives a bijection 0 0 {α ∈ C|αΛ ⊂ Λ } ↔ {nontrivial holomorphic maps φ : C/Λ → C/Λ with φ(0) = 0}. Proof. See [Sil86, Chapter VI, Theorem 4.1]. We have two immediate corollaries to Theroem 3.1. Corollary 3.2. Every isogeny between elliptic curves ϕ : E → E0 is given analytically by a map of the form 0 ϕ : C/Λ → C/Λ z 7→ αz for some α ∈ C. Corollary 3.3. Two elliptic curves are isomorphic as algebraic varieties if and only if they are isomorphic as analytic manifolds. Finally, Theorem 3.1 combined with some analytic results on the j-invariant gives the following:

Corollary 3.4. Two elliptic curves E and E0 are isomorphic over C if and only if j(E) = j(E0). Proof. Since j(Λ) = j(Λ0) whenever Λ = αΛ0, we can deﬁne j as a function on L. Associating each element of L with a point on H/SL2(Z), we can consider j as a map from H/SL2(Z) to C. Proposition 5 of Chapter VII of Serre [Ser73] shows AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 9

0 that j deﬁnes a bijection in this way between H/SL2(Z) and C. So j(E) = j(E ) if and only if j(Λ) = j(Λ0) if and only if Λ and Λ0 are homothetic, if and only if E 0 and E are isomorphic (Corollary 3.3 and Claim 2.1).

0 0 Note that Corollary 3.4 implies that j(τ) = j(τ ) if and only if τ ≡ τ mod SL2(Z).

0 We have seen in Section 2 a bijection between H/Γ0(N) and triplets E,E , ϕ with ϕ : E → E0 where ϕ is an analytic map and ker(ϕ) is cyclic order N. We can now deﬁne a bijection between the later set and pairs of lattices (Λ, Λ0) such that Λ is a cyclic sublattice of Λ0 of order N, i.e. Λ ⊂ Λ0 and Λ0/Λ is a cyclic group of order N. We consider two pairs (Λ, Λ0) and (L, L0) to be equivalent if there is an α ∈ C such that L = αΛ and L0 = αΛ0. Given ϕ : E → E0, with E = C/Λ and E0 = C/Λ0, Theorem 3.1 says that there exists α ∈ C such that ϕ is the map sending z ∈ C/Λ to αz. Thus we have αΛ ⊂ Λ0. Since Λ and αΛ are homothetic, we have E =∼ C/αΛ, so we can assume without loss of generality that Λ ⊂ Λ0 and ϕ sends z + Λ to z + Λ0. Then ker(ϕ) = Λ0/Λ is a cyclic group of order N so Λ is a cyclic sublattice of Λ0 of order N. Given a pair (Λ, Λ0) such that Λ is a cyclic sublattice of Λ0 of order N, the map ϕ : C/Λ → C/Λ0 has kernel Λ0/Λ which is a cyclic subgroup of order N, so letting 0 0 0 E = C/Λ and E = C/Λ gives our triplet E,E , ϕ . Clearly given a pair (Λ, Λ0), an equivalent pair (αΛ, αΛ0) will give rise to an 0 0 0 equivalent triplet. If E,E , ϕ gives the pair Λ, Λ ) and E0,E0, ϕ0 is an equiv- 0 0 ∼ 0 0 0 alent triplet to E,E , ϕ , then E0 = E so there exists α ∈ C such that Λ0 = αΛ . ∼ ∼ 0 0 Then E0 = E = Λ/αΛ and αΛ ⊂ Λ0 so we have deﬁned the pair of lattices (Λ0, Λ0) 0 0 arising from the triplet E0,E0, ϕ0 to be (αΛ, αΛ ). This completes the argument that the set of equivalence classes of triplets E,E0, ϕ with ϕ : E → E0 and ker(ϕ) cyclic order N is in bijection with the set of equivalence classes of pairs of lattices (Λ, Λ0) such that Λ is a cyclic sublattice of Λ0 of order N.

We have deﬁned X0(N) analytically in the previous section; in fact X0(N) also has the structure of a projective variety as follows. For each positive integer N, let {γi} be a set of representatives of the right cosets of Γ0(N) in SL2(Z). We deﬁne a function ΦN of two complex variables as follows Y ΦbN (X, τ) := (X − j(Nγiτ)).

Since j is invariant under the action of SL2(Z), it does not matter which coset representatives we choose. We can show analytically that the coeﬃcients of X in ΦN are polynomials in j(τ) with integer coeﬃcients, hence we have a polynomial Q ΦN (X,Y ) ∈ Z[X,Y ] such that ΦN (X, j(τ)) = (X − j(Nγiτ)) [Cox89, Chapter 11]. The locus of the polynomial ΦN (X,Y ) is a projective variety in P2(C) that is isomorphic analytically to the closure of X0(N). Indeed, we have the following proposition:

Proposition 3.5. If u, v ∈ C then ΦN (u, v) = 0 if and only if there exist lattices L and L0 such that [L : L0] = N, L/L0 is cyclic, u = j(L), and v = j(L0).

Proof. See [Cox89, Theorem 11.23]. 10 SHELLY MANBER

Recall that the points of H/Γ0(N) correspond bijectively to pairs of lattices (L, L0) such that [L : L0] = N and L/L0 is cyclic, and that the correspondence is τ given explicitly by τ 7→ (h N , 1i, hτ, 1i) for τ ∈ H/Γ0(N). Let ϑ : H/Γ0(N) → X0(N) be deﬁned as follows:

ϑ : H/Γ0(N) → X0(N) τ τ 7→ j( ), j(τ) . N

Then Proposition 3.5 shows that ϑ is bijective. Since j is holomorphic on H, ϑ deﬁnes an isomorphism of analytic manifolds. Notice that Φb1(X, τ) = X − j(τ), so Φ1(X,Y ) = X − Y . Indeed, we have mentioned already in the proof of 3.4 that j deﬁnes a bijection between H/SL2(Z) and C.

4. Complex Multiplication

Our ultimate goal is to produce an explicit set of points on X0(N) that form a complete set of Galois conjugates in a finite abelian extension of Q. In order to achieve this, we will first construct such a set on X0, then generalize to X0(N). Class field theory tells us that given an order O in an imaginary quadratic number field K, the elements of the ideal class group C(O), i.e. the Picard group of O, are naturally identified with the Galois group of the maximal unramified abelian extension H of O over K such that the group action of an element of C(O) is the same as the Galois action of it’s image in Gal H/K. The goal of this section is to connect the ideal class group of an order in an imaginary quadratic number field to a set of elliptic curves. This is accomplished by the theory of complex multiplication. Later, we will see that the images of this set of curves in X0 lie in the maximal unramified abelian extension of the order. Since a group acts simply and transitively on itself, and the ideal class group action is the same as the action of the galois group, we will have shown that the points on X0 arising from this set of curves is a complete set of galois conjugates. For now, we introduce the notion of elliptic curves with complex multiplication. Let us consider the set of endomorhpisms of an elliptic curve E. We have seen already that all isogenies between elliptic curves correspond analytically to multiplication by a complex number. Thus if E is represented analytically as C/Λ, each endomorphism of E corresponds to a complex number α ∈ C such that αΛ ⊂ Λ. Clearly nΛ ⊂ Λ for all n ∈ Z and for all lattices Λ. For most elliptic curves this is the whole story, i.e. there are no α ∈ C\Z such that αΛ ⊂ Λ. If there are α ∈ C\Z such that αΛ ⊂ Λ, then we say that E has complex multiplication.

Theorem 4.1. If E = C/Λ has complex multiplication then (1) Up to homothety, Λ can be written hτ, 1i such that τ lies in an imaginary quadratic number ﬁeld K (2) Writing Λ in this way, the set of α such that αΛ ⊂ Λ forms a subring O of K. AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 11

Proof. Applying a homothety if needed, write Λ = hτ, 1i and let α ∈ C such that αΛ ⊂ Λ. Then we have α = aτ + b ατ = cτ + d for some integers a, b, c, and d. Substituting for α, we get (aτ + b)τ = cτ + d, and simplifying this yields aτ 2 + (b − c)τ − d = 0, so τ is the solution to a quadratic equation. Since τ ∈ H, it is not real and thus we must have τ ∈ K for some imaginary quadratic number field K. To prove part (2), it is easy to check explicitly that if α1 and α2 are such that α1Λ ⊂ Λ and α2Λ ⊂ Λ, then the same holds true for α1α2 and α1 + α2. Thus O is a ring. Finally, we have Λ = hτ, 1i so if β ∈ C such that βΛ ⊂ Λ then in particular β ∗ 1 ∈ Λ ⊂ K. So O is a subring of K. Using the notation of the above theorem, we say that O is the endomorphism ring of E and that E has complex multiplication by O, or that E has CM by O. Note that when we say that E has CM by O, we mean that O is isomorphic to the full endomorphism ring of E, and not merely that each element of O preserves the lattice. Thus for each elliptic curve E, there is exactly one ring O up to isomorphism of which we can say, “E has CM by O.” Definition 2. An order O in an imaginary quadratic number field K is a subring of K such that 1 ∈ O, O is a finitely generated Z-module, and O contains a basis for K as a Q vector space.

All orders in K are contained in the ring of integers OK of K, and OK is always an order of K. For this reason, we call OK the maximal order of K. Furthermore, we will see in the next section that all orders in K are lattices in C. Theorem 4.2. If E = C/Λ has complex multiplication by O ⊂ K then O is an order in K.

Proof. See [Sil86, Chapter 6, Theorem 5.5]. Let O be an order in an imaginary quadratic number field and let M be a finitely- generated O-module. We say that M is invertible if there exists an O-module M −1 −1 ∼ such that M ⊗O M = O. Equivalently, a finitely generated O-module is invertible if it is locally free of rank one, i.e. if Mp is isomorphic to Op as an Op-module for all primes p of O. The set of invertible O-modules up to isomorphism is a finite group called the Picard group, with group action given by tensor product and O as the identity element. The Picard group of O is denoted Pic O. If O is a domain whose fraction field is an algebraic extension L of Q then a finitely generated O-submodule of L is called a fractional ideal of O. Since fractional ideals are subsets of L, we can multiply two fractional ideals by multiplying elementwise. In other words, for fractional ideals I and J of the same ring O we define IJ to be the ring generated by the set {ij | i ∈ I, j ∈ J}. A fractional ideal J is invertible if there exists a fractional ideal J −1 such that JJ −1 = O, and if this is the case then J −1 can be given explicitly as J −1 = {x ∈ L | xJ ⊂ O}. The set of 12 SHELLY MANBER invertible fractional ideals of O, denoted I(O), is a group under the operation of ideal multiplication. A principal fractional ideal is a fractional ideal of the form xO for some x ∈ L. The set of invertible principal fractional ideals P(O) forms a subgroup of I(O), and C(O) = I(O)/P(O) is a finite group called the ideal class group that is isomorphic to Pic O.

Theorem 4.3. If O is an order in an imaginary quadratic number ﬁeld K and C/Λ has complex multiplication by O then Λ is an invertible O-module.

Proof. First note that Λ is isomorphic to an O-module of the form hτ, 1i, so we can assume without loss of generality that Λ = hτ, 1i ⊂ K. It remains to show that Λ is a locally free O-module of rank one. Let α ∈ O \ Z, or in other words αΛ ⊂ Λ and α∈ / Z. Then there exist integers a, b, c, and d such that ατ = aτ + b α = cτ + d with b 6= 0 and c 6= 0. We can assume without loss of generality that α is indivisible in O, i.e. there does not exist an integer m such that α = mα0 with α0 ∈ O (otherwise choose α0 instead of α). Let p be a prime ideal of O and p the prime integer such that p ∩ Z = (p). Since Λ is imbedded in K, we can describe Λp explicitly by Λp = {λ/y | λ ∈ Λ, y ∈ O \ p} = ΛOp. We split the proof into four cases.

Case 1: c 6≡ 0 mod p. α−d Then c∈ / p so τ = c ∈ Op. Thus Λp = hτ, 1iOp ⊂ Op. Obviously Op ⊂ hτ, 1iOp, so Λp = Op.

Case 2: b 6≡ 0 mod p. ατ−aτ α−a Then b∈ / p so 1 = b = b τ ∈ τOp, so Λp = hτ, 1iOp ⊂ τOp. Obviously τOp ⊂ hτ, 1iOp, so Λp = τOp. Thus Λp is free Op-module generated by τ and is isomorphic as an Op-module to Op.

Case 3: c ≡ 0 mod p, b ≡ 0 mod p and d − a 6≡ 0 mod p. We have

α(τ + 1) = ατ + α = aτ + b + cτ + d = (a + c)(τ + 1) + b + d − a − c

. Since Λ = hτ, 1i = hτ + 1, 1i we can replace τ by τ + 1. Then b + d − a − c ≡ d − a mod p and d − a 6≡ 0 mod p, so we can use the analysis in case 2 to show that Λ is locally free.

Case 4: c ≡ 0 mod p, b ≡ 0 mod p and d − a ≡ 0 mod p. 2 ep Recall that cτ + (d − a)τ − b = 0. Let ep be the largest integer such that p divides b, c, and d−a, and chose any integers x and y such that x−y = (d−a)/pep ∈ AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 13

. Now let Z c α0 = τ + x, pep then c α0τ = τ 2 + xτ pep 1 = (b − (d − a)τ) + xτ pep b = yτ + , pep c b and we have pep , pep , x, y ∈ Z. Furthermore, our choice of ep guarantees that p c b 0 does not divide pep , pep and x − y ≡ simultaneously. Thus replacing α with α , one of the above three cases will prove that Λ is locally free. The above theorem tells us that if E = C/Λ is an elliptic curve with CM by O then Λ is an invertible fractional ideal of O. Let Ell(O) denote the set of isomorphism classes of elliptic curves with complex multiplication by O. Then in fact, we have the following explicit connection between Ell(O) and C(O). Theorem 4.4. The set of elliptic curves (up to isomorphism) with complex multi- plcation by O is in natural bijection with the elements of C(O) =∼ Pic O.

Proof. Let E = C/Λ be an elliptic curve with CM by O. Then Λ is an invertible fractional ideal of O, and by deﬁnition a lattice Λ0 lies in the same ideal class as Λ 0 if and only if there exists x ∈ K such that Λ = xΛ. Let µ0 : Ell(O) → I(O) be the map that sends E = C/Λ to Λ. We saw in Section 2 that if E0 = C/Λ0 then E =∼ E0 if and only if Λ0 = xΛ for some x ∈ K, i.e. if and only if Λ and Λ0 belong to the same ideal class. Thus µ0 induces an injective map µ : Ell(O) → C(O) C/Λ 7→ Λ. Now let α ∈ C(O), and let J be an invertible fractional ideal that represents α. If x ∈ J −1, then xJ ⊂ O and xJ also represents α. Since O, is a lattice in C and xJ an additive subgroup of O, we must have xJ = {mλ1 + nλ2 | m, n ∈ Z} for some λ1, λ2 ∈ O. To show that xJ is a lattice in C, all that remains is to show that λ1/λ2 ∈/ R, for then xJ = hλ1, λ2i with either λ1/λ2 ∈ H or λ2/λ1 ∈ H. If λ1/λ2 ∈ R, then every element of xJ is of the form rλ2 with r ∈ R, and hence the ratio of any two elements of xJ is in R. Let j ∈ J and ω ∈ O \R. Then xj ∈ xJ and xjτ ∈ xJ but xjτ/xj = τ∈ / R. Thus λ1/λ2 ∈/ R, and xJ is a lattice in C. Now µ(C/xJ) = α and hence µ is surjective. So µ deﬁnes a bijection between Ell(O) and C(O).

5. Orders in Imaginary Quadratic Number Fields In this section, we will present some facts about orders in imaginary quadratic number fields. Most of the results have very nice and elementary proofs in [Cox89], so we will refer the reader to the proofs there. Throughout this section, we let O denote an order in an imaginary quadratic number field K of discriminant D. We denote the ring of integers of K by OK . For an imaginary qudaratic number field, we have OK = Z + Z[ω] where 14 SHELLY MANBER

( √ 1+ D if D ≡ 1 mod 4 ω = √ 2 D otherwise.

Claim 5.1. There exists a positive integer f such that O = Z + fOK = Z + Z[fω].

Proof. Since O is a subring of OK , we have 1 ∈ O so Z ∈ O. Since O contains a Q-basis for K, we must have α ∈ O for some α ∈ OK \ Z. Then α = a + bω for some a, b ∈ Z, so bω ∈ O. Let f be the smallest integer such that fω ∈ O. Then O = Z + Z[fω], for if c + dω ∈ O with f 6≡ 0 mod d then gcd(d, f)ω ∈ O and gcd(d, f) < f, which is impossible by minimality of f.

The conductor of an order O is the index of O in its maximal order [OK : O]. Claim 5.1 implies that all orders have ﬁnite conductor and are uniquely determined by their conductor. Indeed, every order O is of the form Z + fOK where f is the conductor of O. Claim 5.1 also implies that all orders O are lattices in C. The ideal class group of OK is well understood: since OK is a Dedekind domain, all fractional ideals of OK are invertible so I(OK ) is the set of all fractional ideals of OK and P(OK ) is the set of all principal fractional ideals of OK . Deﬁnition 3. If J is a fractional ideal of O then the norm of J, denoted N (J) is the lattice index [O : J] of J in O. Proposition 5.2. We have the following facts concerning norms of fractional ideals.

(1) If α ∈ O then N (αO) = NK/Q(α) where NK/Q(α) is the norm of α as an element over Q. (2) N (JJ 0) = N (J)N (J 0)

Proof. See [Ste04, Section 10.3] or [Cox89, Lemma 7.14]. Theorem 5.3. If J ⊂ J 0 and I are invertible fractional ideals of O, then [IJ 0 : IJ] = [J 0 : J]. Proof. By Lemma 2.4, [O : IJ 0][IJ 0 : IJ] = [O : IJ], so (5) N (IJ 0)[IJ 0 : IJ] = N (IJ). Similarly, [O : J 0][J 0 : J] = [O : J] and hence (6) N (J 0)[J 0 : J] = N (J). By Proposition 5.2, (5) gives us (7) N (I)N (J 0)[IJ 0 : IJ] = N (I)N (J). Then (6) and (7) give [IJ 0 : IJ] = N (J)/N (J 0) = [J 0 : J]. We say that a fractional ideal J of O is prime to f if the norm of J is relatively prime to f. Let I(O, f) denote the set of all fractional ideals of O that are prime to f and P(O, f) the set of fractional ideals in I(O, f) that are principal. Claim 5.4. We have C(O) =∼ I(O, f)/P(O, f)

Proof. See [Cox89, Proposition 7.19]. AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 15

This is a fairly explicit description of C(O), but we can do better; we can describe C(O) in terms of the ideals of OK . Theorem 5.5. For an order O of K of conductor f, the map

ϕ : I(OK , f) → I(O, f) J 7→ J ∩ O is an isomorphism, with inverse map −1 ϕ : I(O, f) → I(OK , f)

J 7→ JOK .

Furthermore, ϕ preserves norm, i.e. the norm of an ideal J prime to f in OK is equal to the norm of ϕ(J) in O.

Proof. See [Cox89, Proposition 7.20].

Let PZ(OK , f) be the subset of P(OK , f) whose generator is an element of O. An equivalent condition to define PZ(OK , f) is the set of principal ideals (α) of OK such that α ≡ a mod fOK with a ∈ Z and gcd(a, f) = 1. ∼ Theorem 5.6. We have C(O) = I(O, f)/P(O, f) = I(OK , f)/PZ(OK , f). Proof. See [Cox89, Proposition 7.22]. 6. Class Field Theory Let O be an order in an imaginary quadratic number field K and, as in Section 4, denote by Ell(O) the set of elliptic curves up to isomorphism with complex multiplication by O. We have seen in Section 4 that Ell(O) is a finite set in bijection with the elements of C(O). We also have from Section 2 that the elements of Ell(O) ⊂ L are in bijection with a set of points in H/SL2(Z). Let CM(O) := {τ ∈ H/SL2(Z) | Oτ = O}, where Oτ is defined to be the endomorphism ring of C/hτ, 1i, or in other words, Oτ := {α ∈ C | αhτ, 1i ⊂ hτ, 1i}. Now we have explicit bijections between the four finite sets Pic O, C(O), Ell(O), and CM(O). For τ ∈ CM(O), the corresponding element of Ell(O) is the elliptic curve C/hτ, 1i, the corresponding element of C(O) is the ideal class of the fractional ideal hτ, 1i of O, and the corresponding element of Pic O is the isomorphism class of hτ, 1i as an O-module. Let ν : CM(O) → Pic O denote this last bijection. Now Pic O acts simply and transitively on itself by the group action, thus it acts simply and transitively on Ell(O) via the identification between Ell(O) and Pic O. More formally, we define a simply transitive action ∗ of Pic O on Ell(O) as follows: for α ∈ Pic O and E ∈ Ell(O) let α ∗ E := µ−1(α−1 ⊗ µ(E)). The equivalent action of Pic O on CM(O), namely α ? τ := ν−1(α−1 ⊗ ν(τ)), takes τ to an element ω ∈ CM(O) such that hω, 1i is homothetic to I−1hτ, 1i for some I representing α. At the end of this section, we will prove that for each α ∈ Pic O, there exists an I representing α such that I−1hτ, 1i = hω, 1i for some ω ∈ H. 16 SHELLY MANBER

Yet another bijection that we have established is that between elements of H/SL2(Z) and points on the algebraic curve X0. Let Υ(O) denote the image of CM(O) on X0 via this bijection (i.e. Υ(O) = ϑ(CM(O))); then we have an induced action of Pic O on Υ(O). We claim that the points in Υ(O) are defined over a finite algebraic extension H of K; we will then identify the action of Pic O with the action of Gal H/K. The first step is to show that if τ ∈ CM(O) then j(τ) is an algebraic integer. To this end, we examine the action of Aut(C/Q) on the j-invariants of elliptic curves. If E is an elliptic curve, then Aut(C/Q) acts on E by acting on the coefficients of the defining algebraic equation; let us denote the action of σ ∈ Aut(C/Q) on E by Eσ. Since the j-invariant of an elliptic curve is a polynomial in its coefficients, the j invariant of Eσ is σ(j(E)).

Claim 6.1. For all τ ∈ CM(O), j(τ) is algebraic over Q. Proof. (expanded from [Dar04, Theorem 5.2]) The goal is to show that the set {σ(j(τ)) | σ ∈ Aut(C/Q)} is ﬁnite, since a trancendental element has an inﬁnite orbit under Aut(C/Q). Let τ ∈ CM(O), Eτ = C/hτ, 1i, and σ ∈ Aut(C/Q), and consider the map

σ θσ : End Eτ → End Eτ ϕ 7→ σ ◦ ϕ ◦ σ−1.

It is easy to check that θσ deﬁnes a group homomorphism. We can deﬁne an inverse of θσ by

−1 σ θσ : End Eτ → End Eτ ϕ 7→ σ−1 ◦ ϕ ◦ σ,

σ ∼ σ so θσ is an isomorphism and End Eτ = O. Thus the set {Eτ | σ ∈ Aut(C/Q)} is a subset of {Eτ | τ ∈ CM(O)}, and in particular is ﬁnite. Since j(Eτ ) = j(τ) and σ j(Eτ ) = σ(j(Eτ )) = σ(j(τ)), the set {σ(j(τ)) | σ ∈ Aut(C/Q)} is ﬁnite and j(τ) is algebraic.

Let Q denote the algebraic closure of Q and K the algebraic closure of K. Since ϑ : H/SL2(Z) → X0 is given by τ 7→ (j(τ), j(τ)), the coordinates of the elements of Υ(O) lie in Q ⊂ K so there is a well deﬁned Galois action of Gal K/K on Υ(O). The bijection between Υ(O) and CM(O) induces an action of Gal K/K on CM(O), which is equivalent to the standard action of Gal K/K on E ∈ CM(O) since every elliptic curve E can be expressed in such a way that the coeﬃcients are rational functions of j(E). As with Aut(C/Q), we denote the action of σ ∈ Gal K/K on an elliptic curve E by Eσ, and we have that j(Eσ) = σ(j(E)) and End Eσ = End E for all σ ∈ Gal K/K. Let E be an elliptic curve with CM by O. We construct a map η from Gal K/K to Pic O by mapping σ to the unique element of Pic O that takes E to Eσ (recall that the action of Pic O on Ell(O) is simply transitive). In other words, we have η : Gal K/K → Pic O such that Eσ = η(σ) ∗ E.

Claim 6.2. The map η : Gal K/K → Pic O deﬁned above is a homomorphism. AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 17

Proof. Let σ1, σ2 ∈ Gal K/K. Then η(σ1σ2) is the unique element of Pic O such (σ1σ2) that E = η(σ1σ2) ∗ E. Now E(σ1σ2) = (Eσ2 )σ1

σ1 = (η(σ2) ∗ E)

= η(σ1) ∗ (η(σ2) ∗ E)

= (η(σ1)η(σ2)) ∗ E.

So η(σ1σ2) = η(σ1)η(σ2). Claim 6.3. The actions of Pic O and Gal K/K on Ell(O) commute. Proof. The result is proved in [Sil94, Chapter II, Proposition 2.5] for maximal orders and follows also for any order. Claim 6.4. The map η : Gal K/K → Pic O deﬁned above is independent of the choice of E. Proof. Let E0 ∈ Ell(O) such that E0 6= E. Then E0 = γ ∗ E for some γ ∈ Pic O. Let σ ∈ Gal K/K. By deﬁnition, Eσ = η(σ) ∗ E, so (E0)σ = (γ ∗ E)σ = γ ∗ (Eσ) (by Claim 6.3) = γ ∗ (η(σ) ∗ E) = (γ ∗ η(σ)) ∗ E = (η(σ) ∗ γ) ∗ E (since the Picard group is commutative) = η(σ) ∗ (γ ∗ E) = η(σ) ∗ E0

ker(η) Now let H := K , so Gal H/K =∼ (Gal K/K)/ker(η) =∼ η(Gal K/K) is a subgroup of Pic O. In particular, it is finite and abelian. Note that j(E) ∈ H for all E ∈ Ell(O). Indeed, if σ ∈ ker(η) then Eσ =∼ E hence j(E) = j(Eσ), and, as we have seen above, j(Eσ) = σ(j(E)), so j(E) is fixed by all σ ∈ ker(η). The next part of this section constructs a homomorphism φ : Pic O → Gal L/K where L is a finite, abelian, Galois extension of K. We will see later that H = L and these maps are inverses.

Theorem 6.5. Let L be a Galois extension of K, p a prime of OK that is unram- iﬁed in L, and P a prime of OL lying above p. Then there is a unique element σ ∈ Gal L/K such that σ(α) = αN(p) mod P L/K where N(p) = OK /p is the norm of p. We denote this element σ by P .

Proof. See [Cox89, Lemma 5.19].

L/K L/K −1 Theorem 6.6. If σ ∈ Gal L/K then σ(P) = σ P σ .

Proof. See [Cox89, Corollary 5.21]. 18 SHELLY MANBER

L/K Theorem 6.6 implies that if L is an abelian extension of K then P does not L/K depend on the prime ideal P of OL lying above p. In this case, we denote P L/K simply as p . Let L be an abelian unramified Galois extension of K. Then we define a map φ : I(OK ) → Gal L/K called the Artin map as follows: for each J ∈ I(OK ) let Q pep be the unique factorization of J in terms of prime ideals (so only finitely p∈OK many ep are non-zero). Then e Y L/K p φ(J) := , p p∈I(OK ) p is prime where the product above refers to the group action in Gal L/K (composition). Theorem 6.7. (The Existence Theorem) Let P1(OK , f) denote the subset of principal fractional ideals in P(OK , f) generated by α ∈ K such that α ≡ 1 mod fOK , and let G be a subgroup of I(OK , f) containing P1(OK , f). Then there exists a unique Abelian extension L of K such that all primes of OK that are prime to f are unramified in L and G is the kernel of the Artin map φ : I(OK , f) → Gal L/K.

Proof. See [Cox89, Theorem 8.6].

For each G, the Artin map induces an isomorphism I(OK , f)/G → Gal L/K; by abuse of notation we will denote this map by φ as well. Corollary 6.8. If O is an order in K of conductor f then there exists a unique Abelian extension L and K such that φ : C(O) →∼ Gal L/K.

Proof. Let G = PZ(OK , f) ⊃ P1(OK , f). Then Theorem 6.7 tells us that there is ∼ a unique abelian extension L of K such that φ : I(OK , f)/P (OK , f) → Gal L/K ∼ Z and by Theorem 5.6 we have C(O) = I(OK , f)/PZ(OK , f). We call the L of Corollary 6.8 the ring class ﬁeld of O. We are now ready to deﬁne the explicit connection between the actions of Pic O and Gal L/K on Ell(O): Theorem 6.9. H = L and for all α ∈ Pic O, we have η(φ(α)) = α.

Proof. See [Dar04, Proposition 3.4]. Corollary 6.10. For all τ ∈ CM(O) and α ∈ Pic O, we have j(α?τ) = φ(α)(j(τ)).

Proof. Let E = C/hτ, 1i and recall that we have deﬁned the action ? of α on CM(O) as the unique element of H/SL2(Z) such that α ∗ E = C/hα ? τ, 1i. Then Theorem 6.9 gives Eφ(α) = η(φ(α)) ∗ E = α ∗ E,

φ(α) and hence j(α ? τ) = j(α ∗ E) = j(E ) = φ(α)(j(E)) = φ(α)(j(τ)). AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 19

All that remains now is to demonstrate as promised that there is a representative I for each α ∈ C(O) such that 1 is a basis element of I−1hτ, 1i for all τ ∈ CM(O). To do so, we will examine the splitting of primes in an imaginary quadratic extension K of Q. Let p be a prime ideal of OK with p ∩ Z = (p) and p unramified in OK . Since [K : Q] = 2, there are only two possibilities for the splitting of pOK : either pOK = p is prime in OK , or pOK = pp. In the former case, the residue ∼ 2 degree is 2, so OK /p = Fp2 , N (p) = p , and the Frobenius element associated to p ∼ is nontrivial. In the later case, the residue degree is 1, so OK /p = Fp, N (p) = p, the Frobenius element associated to p is the identity. Since K/Q is an abelian extension, Chebotarev’s density theorem tells us that the Dirichlet density of primes p ∈ Z whose Frobenius element is the identity is 1/[K : Q] = 1/2. In other words, if S is the set of primes p that split in K then P 1/ps 1 (8) lim p∈S = . s→1+ − log(s − 1) 2 For each prime p that splits, there are two primes p and p, that lie above p, each having norm p. Thus the Dirichlet density in K of primes p of OK lying above primes p in Z that split completely, or equivalently primes p of OK whose norm is prime in Z, is 2 P 1/ps (9) lim p∈S = 1. s→1+ − log(s − 1) By applying Chebotarev’s density theorem to the Abelian extension L/K where L is the ring class field of O, we see that for each element σ of Gal L/K, the L/K Dirichlet density of primes P of OK with P = σ is 1/[L : K]. Since the number of prime ideals of OK with norm not prime to the conductor is finite, the L/K Dirichlet density of primes P of OK that are prime to f with P = σ is also 1/[L : K]. By Corollary 6.8, this is equivalent to saying that the set of prime ideals in I(OK , f) representing any given equivalence class of I(OK , f)/PZ(OK , f) is of density 1/[L : K].

Let α ∈ I(OK , f)/PZ(OK , f); we claim that the set Ssplit consisting of prime ideals in I(OK , f) representing α that split and therefore have prime norm has a positive Dirichlet density. Indeed, the set Sall consisting of all primes representing α is the union of Ssplit and Sinert. From (9) we have that Sinert has zero density. But we have also seen that Sall has density 1/[L : K]. Therefore Ssplit must have positive ∼ density in OK . Since we have an isomorphism I(OK , f)/PZ(OK , f) → C(O) that preserves norm, this means that for each α ∈ C(O), the set of prime ideals of O representing α that have prime norm has a positive Dirichlet density. We now return to the claim that if α ∈ C(O), there exists a prime ideal I of O such that the lattice I−1hτ, 1i can be expressed in the form hω, 1i. Let J be a prime ideal of O of prime norm p and note that J −1 = {a ∈ K | aJ ⊂ O}, so we must have 1 ∈ J −1 and consequently 1 ∈ J −1hτ, 1i. If we can show that 1/n∈ / J −1hτ, 1i for all integers n > 1, we have seen in the proof of Claim 2.2 that 1 can be extended to a basis hω, 1i of J −1hτ, 1i. Assume to the contrary that 1/n ∈ J −1hτ, 1i for some integer n. If p is relatively prime to n, then there exist integers a, b ∈ Z such that an + bp = 1, so 1 bp = + a. n n 20 SHELLY MANBER

−1 −1 Since 1/n ∈ J hτ, 1i, we have 1/n = j1 + j2τ for some j1, j2 ∈ J , so 1 = bp(j + j τ) + a. n 1 2 −1 Since p is the norm of J, we have pO ⊂ J so pJ ⊂ O. Thus pj1, pj2 ∈ O and 1/n ∈ Ohτ, 1i = hτ, 1i. But this is impossible, as 1 is a basis element of hτ, 1i. So p must divide n, which means that 1/p ∈ J −1hτ, 1i. Since J −1hτ, 1i is an O module, we have 1 (pO)−1 = O ⊂ J −1hτ, 1i. p Since pO = JJ, we have −1 J −1J ⊂ J −1hτ, 1i, and hence −1 J ⊂ hτ, 1i, −1 so J is in the prime decomposition of the fractional O-ideal hτ, 1i. There are only ﬁnitely many primes in the decomposition of hτ, 1i and inﬁnitely many primes J of O representing α with prime norm. Hence, there exists some I representing α such −1 that I is not in the prime decomposition of hτ, 1i. We have shown that there are no elements of the form 1/n in I, so there exists ω such that hω, 1i = I−1hτ, 1i.

7. Heegner Points on X0(N)

Let τ ∈ H ∩K and Oτ be the CM ring of C/hτ, 1i. We denote the ring class field ∼ of Oτ by Hτ and the corresponding Artin map by φτ , i.e. φτ : Pic Oτ → Gal Hτ /K. We have seen in the previous section (Corollary 6.10) that we can define an action ? of Pic Oτ on CM(Oτ ) such that j(α ? ω) = φτ (α)j(ω) for all α ∈ Pic Oτ and all ω ∈ CM(Oτ ). Denoting reduction mod SL2(Z) with overline, define

Sτ = {α ? τ | α ∈ Pic Oτ } ⊂ H/SL2(Z).

Then we have seen that j(s) ∈ Hτ for all s ∈ Sτ and that

{j(s) | s ∈ Sτ } = {φτ (α)j(τ) | α ∈ Pic Oτ }

= {σ(j(τ)) | σ ∈ Gal Hτ /K} . Let O be an order in an imaginary quadratic number ﬁeld and deﬁne

CMN (O) := {τ ∈ H/Γ0(N) | O = Oτ = Oτ/N }.

The main task of this section is to deﬁne an action ?N of Pic O on CMN (O) such that

j(α ?N τ) = φ(α)j(τ), and α ? τ τ j( N ) = φ(α)j( ) N N ∼ for all α ∈ Pic O and all τ ∈ CMN (O), where φ : C(O) → Gal H/K is the Artin map deﬁned in Section 6. AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 21

Recall that

ϑ : H/Γ0(N) → X0(N) τ τ 7→ j( ), j(τ) . N

Thus if we can succeed in deﬁning such an action ?N and explicitly computing the set

{α ?N τ | α ∈ Pic O} then we will have achieved the goal of the last several sections, for we will have

{ϑ(α ?N τ) | α ∈ Pic O} = {φ(α)ϑ(τ) | α ∈ Pic O} is an explicitly-computable complete set of Galois conjugates on X0(N).

0 Claim 7.1. Let τ ∈ CMN (O) and α ∈ Pic O. Then there exists a unique τ ∈ H/Γ0(N) such that 0 τ ≡ α ? τ mod SL2(Z)(10) τ 0 τ ≡ α ? mod SL ( )(11) N N 2 Z

Proof. Recall from Section 2 that each τ ∈ H/Γ0(N) corresponds to the pair of τ elliptic curves C/hτ, 1i → C/h N , 1i or in other words the pair of lattices, hτ, 1i ⊂ τ h N , 1i. As in Section 6, let I be a proper ideal of O that represents α such that O/I 0 is cyclic. We have seen in the previous section that there exist ω, ω ∈ H/SL2(Z) 0 τ given by ω := α ? τ, and ω := α ? N such that I−1hτ, 1i = hω, 1i, and τ I−1h , 1i = hω0, 1i. N −1 −1 τ 0 0 Now I hτ, 1i ⊂ I h N , 1i so hω, 1i ⊂ hω , 1i, which means that ω = aω + b for integers a and b. Since 1 ∈ h1, ωi, we can choose ω such that b = 0, so that ω = aω0. τ Now [h N , 1i : hτ, 1i] = N so Theorem 5.3 gives us τ [hω0, 1i : hω, 1i] = [I−1h , 1i : I−1hτ, 1i] = N, N so haω0, 1i has index N in hω0, 1i. Thus we must have a = N and ω0 = ω/N. Letting τ 0 = ω proves existence. 0 ω τ 0 If ω is such that ω ≡ τ mod SL2(Z) and N ≡ N mod SL2(Z) then this means ω 0 τ 0 that up to homothety, the pairs of lattices hω, 1i ⊂ h N , 1i and hτ , 1i ⊂ h N , 1i are the same, i.e. ω and τ 0 generate the same lattice/cyclic sublattice pair. We have seen that lattice/cyclic sublattice pairs are in bijection with elements of H/Γ0(N) 0 so τ ≡ ω mod Γ0(N).

0 τ 0 0 Since τ ∈ CM(Oτ ) and N ∈ CM(Oτ/N ), we have τ ∈ CMN (O). We deﬁne the 0 action ?N of Pic O on CMN (O) by α?N τ := τ for all α ∈ Pic O and τ ∈ CMN (O). 22 SHELLY MANBER

By Corollary 6.10, together with that fact that j is invariant under the action of SL2(Z), we have

j(α ?N τ) = j(α ? τ) = φ(α)j(τ), and α ? τ τ τ j( N ) = j(α ? ) = φ(α)j( ) N N N as desired.

8. Modularity The Modularity Theorem, once called the Taniyama-Shimura-Weil conjecture, was proved in 2001 by Breuil, Conrad, Diamond, and Taylor. The modularity theorem is a deep theorem, that in particular implies Fermat’s Last Theorem. We will state the result in this section, following very closely the presentation in [Dar06, Section 2.1] Let E be an elliptic curve defined by over Q, and let p be a prime in Z. We say that E has good reduction at p if there exists a polynomial describing E with integer coefficients such that the curve over the finite field Fp defined by reducing the coefficients of this polynomial is non-singular. Otherwise, we say that E has bad reduction at p. Every elliptic curve has bad reduction at only finitely many primes. The conductor N of E is an integer that is a product of the primes at which Q fp E has bad reduction, in other words, N = p where fp = 0 if E has good p∈Z reduction at p and fp 6= 0 otherwise. The exact definition of fp depends on the type of bad reduction and is rather technical, so we omit it here.

Theorem 8.1. (Modularity Theorem) For every elliptic curve E defined over Q of conductor N, there exists a morphism ΦE : X0(N) → E that is defined over Q. To define the morphism, we look at some analytic objects attached to an elliptic curve E.

Deﬁnition 4. For an elliptic curve E deﬁned over Q, the L-series L(E, s) attached to E is given by the following formula:

Y −s 1−2s −1 L(E, s) = (1 − app + δpp ) p∈Z where ( 1 if E has good reduction mod p δp = 0 otherwise. and ( p + 1 − #E(Fp) if E has good reduction mod p ap = 0, 1, or − 1 otherwise (depending on the type of bad reduction).

The L-series of an elliptic curve E converges for all s with Re(s) > 3/2, and in this case we have ∞ X −s L(E, s) = aE(n)n n=1 AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 23 where aE(n) ∈ Z. Deﬁne a complex function f using the coeﬃcients of L(E, s) by ∞ X 2πinz f(z) := aE(n)e . n=1

Then f(z) is a weight 2 modular form for the congruence subgroup Γ0(N). In a b other words, f is meromorphic in the upper half plane, and for every ∈ c d az+b 2 Γ0(N), we have f( cz+d ) = (cz + d) f(z). Then means that ωf := 2πif(z)dz is a az+b 1 differential form on X0(N), since d( cz+d ) = (cz+d)2 dz. Now if we fix τ ∈ H then the set Z γτ { ωf | γ ∈ Γ0(N)} τ is a lattice Λf in C and C/Λf is Q-isogenous to E, i.e. there is an isogeny θ : C/Λf → E that is defined algebraically over Q. Using this, we can define the map we want:

ΦE : H/Γ0(N) → E Z τ τ 7→ θ( ωf ). i∞

ΦE is a well-deﬁned analytic map, for if γ ∈ Γ0(N) then Z τ Z γτ ωf ≡ ωf mod Λf . i∞ i∞ 0 Now the bijection ϑ : H/Γ0(N) ↔ X0(N) induces a map ΦE : X0(N) → E 0 0 such that ΦE = ΦE ◦ ϑ. It turns out that ΦE is a morphism of projective varieties deﬁned over Q. This completes the last piece of our puzzle, for we have already

{ϑ(α ?N τ) | α ∈ Pic O} = {φ(α)ϑ(τ) | α ∈ Pic O} and thus 0 {ΦE(α ?N τ) | α ∈ Pic O} = {ΦE(φ(α)ϑ(τ)) | α ∈ Pic O} 0 = {φ(α)ΦE(ϑ(τ)) | α ∈ Pic O}

= {φ(α)ΦE(τ) | α ∈ Pic O}

= {σ(ΦE(τ)) | σ ∈ Gal H/K} is a set of points on E defined over H invariant under the action of Gal H/K. Since E is an algebraic group, we can sum these points on E. The result will be a fixed point of Gal H/K and thus defined over K. To find a rational point on E, we simply add that point to it’s complex conjugate, for K is quadratic so Gal K/Q is the group consisting of the identity and complex conjugation.

9. Examples Let τ ∈ H∩K be an element of an imaginary quadratic number ﬁeld K satisfying aτ 2 + bτ + c = 0. Claim 9.1. τ is a Heegner point of level N if and only if N|a and gcd(a/N, b, cN) = 1.

Proof. See [Wat05, Proposition 2.2]. 24 SHELLY MANBER

If we let a = N and c = 1, then each choice of b gives us a Heegner point of discriminant D = b2 − 4ac = b2 − 4N. Let us compute a rational point on the elliptic curve y2 + y = x3 − x2 − 10x − 20 of conductor N = 11. If we let b = 3 then we have D = 9 − 4 · 11 = −35. The order of discriminant√ −35 is maximal since 35 is square-free, and the class number of K = Q( −35) is 2, so there is one non-trivial element of the ideal class group of O . Let J ⊂ O be a representative of this element such that O /J is cyclic. K K√ K −3+ −35 2 If we take τ = 22 to be the root of 11x + 3x + 1 in the upper half plane, 0 we can compute a point on E over K by adding the images under ΦE of τ and τ , where τ 0 is chosen such that hτ 0, 1i = J −1hτ, 1i. √ −1+ −35 2 Since −35 ≡ 1 mod 4, we have OK = Z+αZ with α = 2 , so α +α+9 = 0. Then the lattice hα, 3i = 3Z+αZ is a non-principal proper ideal of OK (generated as an OK ideal by α and 3). Let J = hα, 3i, then OK /J is a cyclic group of order −1 α+1 3 and J = h 3 , 1i. We compute that √ √ √ 1 + −35 −3 + −35 25 + −35 J −1hτ, 1i = h , 1ih , 1i = h , 1i 6 22 66 √ 0 25+ −35 so τ = 66 . Using Pari to compute the modular parameterization of E, we ﬁnd that ΦE is given by

ΦE : H/Γ0(N) →E τ 7→(q−2 + 2q−1 + 4 + 5q + 8q2 + q3 + 7q4 − 11q5 + ··· , − q−3 − 3q−2 − 7 ∗ q−1 − 13 − 17 ∗ q − 26 ∗ q2 − 19 ∗ q3 + ··· )

2πiτ 0 where q = e . By computing this series up to 200 terms and adding ΦE(τ ) to ΦE(τ) via the group law on E, we find that √ √ −16 + 2 −35 481 + 148 −35 ( , ) 9 999 is a point on E. In fact, the computer system is only able to compute a decimal value of the form √ √ (−1.7777777... + 1.3146844... −1, 0.481481481... + 0.876456264... −1), √ but since we know that the point lies in Q( −35) and we can compute the decimal value to several hundred places, we can infer that the point is precisely the one listed above. To find a rational point on E, we simply add the point to its complex conjugate, since the Galois group of any imaginary quadratic number field is (1, σ), where 1 is the identity and σ is complex conjugation. We compute that √ √ √ √ −16 + 2 −35 481 + 148 −35 −16 − 2 −35 481 − 148 −35 ( , ) + ( , ) = (5, −6) 9 999 9 999 where “+” denotes addition using the group law on E. It is easily to confirm that indeed (−6)2 − 6 = 53 − 52 − 10 · 5 − 20. Let us now compute another point on the same curve using a non-maximal order.

If we choose b = 4 instead√ of 3 then τ is now the root in the upper half√ plane of 2 −2+ −7 1+ −7 11x + 4x + 1, so τ = . However, the maximal order OK = h , 1i is 11 √ √ 2 not the CM ring of hτ, 1i. Indeed, 1+ −7 τ = −9+ −7 , which is clearly not in hτ, 1i. 2 √22 The CM ring of hτ, 1i is the order O = h1 + −7, 1i of conductor 2 in O ; we √ K √ −9+ −7 √ can check that (1 + −7)τ = 11 = −τ − 1 and (1 + −7) = 11τ + 3. AN ELEMENTARY AND COMPUTATIONAL APPROACH TO HEEGNER POINTS 25

It turns out that the ideal class group of O is trivial, so we would expect that 2πiτ substituting√q = e into the modular parameterization above for E would give a point in Q( −7). Indeed, we get √ √ (−5.5 + 22.488886144... −1, 103 − 44.977772288098... −1) which we can check is very close to the point √ ! −11 + 17 −7 √ , 103 − 17 −7 2 on E. Adding this to it’s complex conjugate gives the integral point (16, −61); again we can check that (−61)2 − 61 = 163 − 162 − 10 ∗ 16 − 20.

10. Acknowledgements and Additional Sources I would like to acknowledge Tomer Schlank, Liang Xiao, Rami Aizenbud, and Gal Binyamini for extensive help in teaching me background material, helping me to understand the articles I read, and coming up with proofs to many of the claim presented here. I would also like to thank my advisers, Joseph Bernstein and Stephen Gelbart. There are several additional sources I used to understand the material but did not cite speciﬁcally in the paper: [For81], [Apo90], [Mil08], [Gro04], [Bir04], [KP05], [Nie02]. Finally, I would like to comment that though the paper incorporates all of the sources and presents the material with some originality of structure, it most heavily draws from and most closely follows the structure of [Dar04].

References [Apo90] Tom Apostol. Modular Functions and Dirichlet Series in Number Theory. Springer- Verlag, 175 Fifth Avenue, New York, New York 10010, USA, 1990. [Bir04] Bryan Birch. Heegner Points: The Beginnings, volume 49. Cam?bridge Univ. Press, 2004. [Cox89] David Cox. Primes of the Form x2 + ny2. John Wiley and Sons, Inc., New York, USA, 1989. [Dar04] Henri Darmon. Rational points on modular elliptic curves, 2004. [Dar06] Henri Darmon. Heegner points, stark-heegner points, and values of l-series. In Proceedings of the International Congress of Mathematicians, volume 2, pages 313–345. European Math. Soc., 2006. [DS05] Fred Diamond and Jerry Shurman. A First Course in Modular Forms. Springer Sci- ence+Business Media, Inc., 233 Spring Street, New York, New York 10013, USA, 2005. [For81] Otto Forster. Lectures on Riemann Surfaces. Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, USA, 1981. [Gro84] Benedict Gross. Heegner Points on X0(N), chapter 5. Ellis Horwood, 1984. [Gro04] Benedict Gross. Heegner points and representation theory, volume 49. Cam?bridge Univ. Press, 2004. [KP05] J˝urgenKl˝unersand Sebastian Pauli. Computing residue class rings and picard groups of arbitrary orders. Journal of Algebra, 2005. [Mil08] James S. Milne. Algebraic number theory (v3.00), 2008. Available at www.jmilne.org/math/. [Nie02] Pace Nielsen. An introduction to orders of number ﬁelds, 2002. [Ser73] J. P. Serre. A Course in Arithmetic. Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, USA, 1973. [Sil86] Joseph Silverman. The Arithmetic of Elliptic Curves. Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, USA, 1986. [Sil94] Joseph Silverman. Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, USA, 1994. 26 SHELLY MANBER

[SS03] Elias Stein and Rami Shakarchi. Complex Analysis. Princeton University Press, 41 William Street, Princeton, New Jersey 08540, 2003. [Ste04] William Stein. A brief introduction to classical and adelic algebraic number theory. available free online, never formally published, 2004. [Wat05] Mark Watkins. Some remarks on Heegner point computations. ArXiv Mathematics e- prints, June 2005.