ALGORITHMS FOR CHOW-HEEGNER POINTS VIA ITERATED INTEGRALS

HENRI DARMON, MICHAEL DAUB, SAM LICHTENSTEIN AND VICTOR ROTGER

(WITH AN APPENDIX BY WILLIAM STEIN)

Abstract. Let E_{/Q} be an elliptic curve of conductor N and let f be the weight 2 newform on Γ_{0}(N) associated to it by modularity. Building on an idea of S. Zhang, the article [DRS] describes the construction of so-called Chow-Heegner points P_{T,f} ∈ E(Q̄) indexed by algebraic correspondences T ⊂ X_{0}(N)×X_{0}(N). It also gives an analytic formula, depending only on the image of T in cohomology under the complex cycle class map, for calculating P_{T,f} numerically via Chen’s theory of iterated integrals. The present work describes an algorithm based on this formula for computing the Chow-Heegner points to arbitrarily high complex accuracy, carries out the computation for all elliptic curves of rank 1 and conductor N < 100 when the cycles T arise from Hecke correspondences, and discusses several important variants of the basic construction.

Contents

1. Introduction 2 Plan of the paper 4 Source code 4 Acknowledgments 4 2. Preliminaries 5 2.1. De Rham cohomology 5 2.2. Iterated integrals 5 2.3. Chow-Heegner points 7 3. Chow-Heegner points on modular curves 8 3.1. Definitions 8 3.2. Algorithms 10 4. Details of the algorithm 12 4.1. Evaluating iterated integrals 13 4.2. Calculating a symplectic basis for H_{dR}^{1}(X/Q)[g] 15 4.3. Modular units and η-products 16 4.4. Computing the Poincaré dual γ_{f} of ω_{f} 17 4.5. Computing the adjustments ∫_{γ_{f}} α 18 4.6. Computing the coefficients c_{ij} 19 4.7. Computing the denominator d_{g,n} 19 4.8. Remarks on complexity 19 5. Numerical examples 22 5.1. Example: 37a1 22 5.2. Example: 43a1 24 5.3. Table 25 5.4. Further observations 25 6. Appendix: Chow-Heegner points associated to pairs of elliptic curves 27 6.1. Introduction 27 6.2. Example: N = 37 28 6.3. The Formula of Yuan-Zhang-Zhang 28 6.4. Iterated Complex Path Integrals 29 6.5. A Numerical Approach to Computing P_{E,F} 29 References 33

1. Introduction

Let E_{/Q} be an elliptic curve of conductor N and let

f = ∑_{n≥1} a_{n}e^{2πinz} ∈ S_{2}(Γ_{0}(N))

be the weight two newform associated to E by modularity. The proof of the Tate conjecture for curves implies the existence of a non-constant modular parametrisation

(1.0.1)  π_{E} : X_{0}(N) ⟶ E

defined over ℚ, together with a regular differential ω ∈ Ω^{1}(E/ℚ) satisfying

(1.0.2)  π_{E}^{∗}(ω) = ω_{f} := 2πif(z)dz.

The map π_{E} plays a key role in the study of the Birch and Swinnerton-Dyer conjecture (BSD). This conjecture predicts that, for any number field H, the rank of E(H) is equal to the order of vanishing r_{an}(E_{/H}) of the L-series L(E_{/H}, s) at s = 1; one is thus led to seek strategies for constructing a non-torsion point P ∈ E(H) when L(E_{/H}, 1) = 0, or—better yet—a system of r_{an}(E_{/H}) independent such points.

When H is a ring class field of an imaginary quadratic field K, a fruitful approach to this question is afforded by the theory of Heegner points attached to K. These are the images under π_{E} of points in X_{0}(N)(K̄) attached to the moduli of elliptic curves with complex multiplication by some order in K. For instance it is known, thanks to the work of Gross– Zagier [GrZa] and its strengthening by Zhang [Zh] combined with a mild generalisation [BD1] of the work of Kolyvagin that Heegner points attached to K can be made to generate all of E(H) ⊗ ℚ when r_{an}(E_{/H}) = [H : K], at least when the discriminant of K is prime to N and L(E_{/K}, s) has a simple zero at s = 1.

Let

ℌ^{∗} = {z ∈ C : Im(z) > 0} ∪ P^{1}(Q) = ℌ ∪ P^{1}(Q)

denote the extended upper-half plane, topologized in the usual manner. The Heegner points attached to K correspond precisely to the quadratic irrationalities z ∈ ℌ ∩ K under the usual identification of X_{0}(N)(ℂ) with Γ_{0}(N)\H^{∗}. Furthermore, after identifying E(ℂ) with the quotient ℂ/Λ_{E} by the period lattice Λ_{E} ⊂ ℂ attached to the regular differential ω of (1.0.2), the map π_{E} can be computed analytically from the formula

(1.0.3)  π_{E}(z) = ∫_{i∞}^{z} ω_{f} = ∑_{n=1}^{∞} \frac{a_{n}}{n} e^{2πinz} ∈ ℂ/Λ_{E},

where the integral is taken over the vertical line joining the cusp i∞ ∈ P^{1}(Q) to z (or any path in Γ_{0}(N)\H^{∗} homotopic to it).

Equation (1.0.3) makes it possible to calculate Heegner points numerically on a computer^{1}, and leads to a efficient algorithm for calculating algebraic points on elliptic curves which is presumably better than the more general-purpose method of descent, at least in situations where it can be made to yield non-torsion points. In addition, a careful study of (1.0.3) has guided attempts to unearth more general analytic recipes for algebraic points on elliptic curves in settings falling outside the scope of the theory of complex multiplication. To elaborate on this last point, let us mention that

1For the important role of experiments in the early study of Heegner points, see for example [Bi], [BG].

• In [BD2] and [Gr], the complex upper half-plane ℌ^{∗} is replaced by its p-adic counterpart H_{p} := ℙ_{1}(ℂ_{p}) − ℙ_{1}(ℚ_{p}) regarded as a rigid analytic variety, and the group Γ_{0}(N) by the group of units in a ℤ[1/p]-order of a totally definite quaternion algebra. Replacing the complex integrals of (1.0.3) by Coleman’s p-adic path integrals leads to p-adic analytic formulae for (and explicit calculations of) Heegner points arising from Shimura curve parametrisations, thanks to the Cerednik-Drinfeld theory of p-adic uniformisation of these curves.

• By mimicking the formulae for Shimura curve parametrisations involving p-adic path integrals, the article [D1] was able to propose a p-adic construction of points on elliptic curves–the so called Stark-Heegner points–conjecturally defined over abelian extensions of real quadratic fields. In this setting, the global structure analogous to (1.0.1) is not currently known to be available, and numerical verifications such as those carried out in [DP] provide the best evidence at present for the conjectures of [D1].

• Let E_{/F} be a modular elliptic curve over a totally real number field F. By calculations similar to those of [D1], but replacing (1.0.3) by a formula involving integrals of complex-analytic differential forms of higher order associated to Hilbert modular forms, [D2, Ch. VIII] proposes a purely complex construction of Stark-Heegner points in E(C). These points are conjectured to be defined over suitable abelian extensions of quadratic extensions K/F which have exactly one non-real archimedean place. In this complex setting as well, numerical experiments such as those described in [DL] are the best evidence so far for the algebraicity of the Stark-Heegner points.

• Complex Stark-Heegner points can be viewed as images of certain null-homologous topological cycles on Hilbert modular varieties under a kind of “Abel-Jacobi map”. In the hope of better understanding this construction, the articles [BDP2] and [DRS] discuss “generalised modular parametrisations”

Π : CH^{r+1}(V)_{0} ⟶ E

from the Chow-group CH^{r+1}(V)_{0} of null-homologous codimension r+1 algebraic cycles on a (2r + 1)-dimensional variety V to the Mordell-Weil group of an elliptic curve E, arising from suitable algebraic correspondences Π ∈ CH^{r+1}(V × E). In the setting of [BDP2], E is an elliptic curve with complex multiplication, V the product of an (r + 1)-dimension Kuga-Sato variety with E^{r}, and the existence of Π relies on an ostensibly difficult instance of the Hodge or Tate conjectures. Nonetheless, the global nature of the points in the image of Π, for which the term “Chow-Heegner point” was proposed in [BDP2], is better understood than in the setting of Stark-Heegner points.

An even simpler instance of the Chow-Heegner point construction arises when V = X_{0}(N)^{3} is the triple product of a modular curve and Π ⊂ X_{0}(N)^{3} × E is the product of a codimension 1 cycle T ⊂ X_{0}(N)^{2} with the graph of the “classical” modular parametrisation π_{E} of equation (1.0.1). The resulting map

Π_{T} : CH^{2}(X_{0}(N)^{3})_{0} ⟶ E,

is a natural generalisation of (1.0.1). In particular the image P_{T,f} := Π_{T} (Δ_{GKS}) of the Gross-Schoen modified diagonal cycle Δ_{GKS} (as defined in Theorem 1 of [DRS] for example) under Π_{T} is an interesting rational point on E whose relations with special values of L-functions [YZZ] and the pro-nilpotent fundamental groups of modular curves [DRS] are well documented in the literature. The point P_{T,f} ∈ E(ℚ) shall be called the Chow-Heegner point attached to the cycle T and the newform f. An alternate, somewhat simpler description of P_{T,f} proceeds by essentially intersecting T with the diagonal in X_{0}(N)×X_{0}(N) to obtain a class in Pic(X_{0}(N)), and hence (after a suitable modification to make it of degree 0) a rational point P_{T} of the Jacobian J_{0}(N). The point P_{T,f} is then just the image of P_{T} under π_{E}. The definition of the points P_{T} and P_{T,f} is described more precisely in §2.3 and §3.1 below.

The article [DRS] also gives a complex analytic formula for the point P_{T,f} in terms of K.- T. Chen’s iterated path integrals. In order to provide some motivation for it, note that the homomorphism

π_{1}(X_{0}(N)(C); ∞) ⟶ C defined by γ ↦ ∫_{γ} ω_{f}

factors through the maximal abelian quotient H_{1}(X_{0}(N)(C), Z) of the fundamental group of X_{0}(N)(ℂ). Chen observed that it is also possible to obtain non-abelian information about this fundamental group through integration. His theory of iterated integrals gives rise to functions

J : π_{1}(X_{0}(N)(C); ∞) ⟶ C

attached to an r-tuple of differentials, which need not factor through H_{1}(X_{0}(N)(C), Z) but rather through the quotient of π_{1}(X_{0}(N)(ℂ), ℤ) by the r-th term in its lower central series. The algorithm for the computation of the Chow-Heegner points P_{T,f} described in this paper, in effect, replaces the integral appearing in (1.0.3) by a sum of iterated integrals attached to pairs of modular forms. These pairs depend only on the so called Hodge cycle attached to T, namely, its image

ξ_{T} ∈ H_{dR}^{1,1}(X_{0}(N)^{2}) ∩ H_{B}^{2}(X_{0}(N)^{2}, ℚ)

under the cycle class map. While algebraic equations for the modular curve X_{0}(N) (and, a fortiori, for the cycle T itself) can be hard to calculate in practice, the Hodge class ξ_{T} is amenable to numerical calculation using the explicit description of the de Rham cohomology of modular curves in terms of weakly holomorphic modular forms of weight two and their q-expansions. The algorithm for computing the Hodge class ξ_{T} and the associated iterated integrals has been implemented in a computer program using the free software package Sage [S^{+}09], and makes the Chow-Heegner points P_{T,f} computable in practice for elliptic curves E of moderate conductor.

As explained in the appendix, there are other, somewhat more direct approaches to computing P_{T,f} for specific cycles T arising from elliptic curve factors of J_{0}(N). The approach based on iterated integrals appears to be somewhat more general and works for arbitrary Hodge cycles. In future work, we hope to extend our algorithm to compute Chow-Heegner points associated to “exceptional” Hodge cycles on products of Kuga-Sato varieties arising from CM forms. The rationality of Chow-Heegner points computed in this manner would provide some indirect numerical evidence for certain open cases of the Hodge conjecture, in the spirit of the study carried out in [BDP2].

Plan of the paper. In §2 we recall necessary facts about iterated integrals and related ingredients for our main algorithm. In §3 we specialize to the case of modular curves, define the points P_{T,f} precisely, and write down an explicit analytic formula for them. In §4 we describe in detail an algorithm for evaluating this formula numerically. The algorithm is illustrated with numerical examples in §5, which also includes a table of triple Chow-Heegner points on rank one elliptic curves of small conductor.

Source code. Our Sage implementation of the algorithm described in this paper can be

found at http://math.berkeley.edu/~mwdaub.

Acknowledgments. We warmly thank the organizers of the 2011 Arizona Winter School for inviting the first and fourth author to deliver a series of lectures at the meeting, for creating such an inspiring atmosphere and giving the four of us the opportunity of working together.

2. Preliminaries

2.1. De Rham cohomology. Let X be a smooth, complete algebraic curve of genus g ≥ 2 over Q, and let Y = X \ {∞} be the complement of a single point in X(Q). For a smooth variety V defined over Q (such as X or Y) we denote by V^{an} the complex manifold V (C) with its analytic topology.

The de Rham cohomology H_{dR}^{1}(X^{an}, C) is the cohomology of the de Rham complex of smooth C-valued differential forms on X^{an}.

Since the Riemann surface X^{an} is algebraic and defined over ℚ, there is a canonical isomorphism

H_{dR}^{1}(X^{an}, C) = H_{dR}^{1}(X/ℚ) ⊗ C.

Here

H_{dR}^{1}(X/Q) := H^{1}(0 → O_{X} → Ω_{X}^{1} → 0)

is the algebraic de Rham cohomology of X/Q, defined as the hypercohomology of the de Rham complex of sheaves of regular differential forms on X.

Because X is a curve, the group H_{dR}^{1}(X/Q) has a particularly simple description in terms of the space Ω_{II}^{1}(X) of differentials of the second kind on X, defined as

Ω_{II}^{1}(X) := rational 1-forms on X with vanishing residues at all points of X.

Observe that Ω_{II}^{1}(X) = Ω_{II}^{1}(Y), by the residue formula. Thus there is a canonical isomorphism

H_{dR}^{1}(X/Q) = Ω_{II}^{1}(Y)/dQ(Y),

where Q(Y) = Q(X) is the field of rational functions on Y (or on X). Using the Riemann-Roch theorem, one sees that

Ω_{II}^{1}(Y)/dQ(Y) ≅ Ω^{1}(Y)/dO(Y).

So H_{dR}^{1}(X/Q) can also be computed as the space of regular 1-forms on Y, modulo exact forms. For computational purposes, the latter description is the most useful: we will compute with classes in H_{dR}^{1}(Y) using rational 1-forms on X which are regular away from the point ∞. These are amenable to computation via their Laurent expansions about ∞.

Later on we shall make use of the symplectic Poincaré pairing

⟨, ⟩ : H_{dR}^{1}(X^{an}, C) × H_{dR}^{1}(X^{an}, C) → C,

defined by

⟨ω, η⟩ := \frac{1}{2πi} ∫_{X} ω ∧ η

for (the cohomology classes of) smooth C-valued 1-forms ω, η on X^{an}. If ω and η are differentials of the second kind on X, regular away from the cusp ∞, then the induced pairing on H_{dR}^{1}(X/Q) can be computed as

⟨ω, η⟩ = res_{∞}(F_{ω} · η) = −res_{∞}(ω · F_{η}),

where F_{ν} denotes a local primitive of the differential ν at ∞.

2.2. Iterated integrals. We now turn to recalling the definition and basic properties of iterated integrals; see [Ch], [H1], [H2] for more details.

Fix a base point o ∈ Y^{an} and denote by Γ := π_{1}(Y^{an}; o) the fundamental group of the Riemann surface Y^{an}. We write I ⊂ Z[Γ] for augmentation ideal of the integral group ring of Γ. Recall that H_{1}(X^{an}, Z) = H_{1}(Y^{an}, Z) ≅ Γ^{ab}, as can be seen from the well-known presentation for the fundamental group of a Riemann surface, and that this abelian group is naturally identified with I/I^{2}.

Definition 2.2.1. The path space on Y based at o, denoted P(Y; o), is the set of piecewisesmooth paths

γ : [0, 1] ⟶ Y^{an}, with γ(0) = o.

Let

π : Ỹ → Y^{an}, resp. π : X̃ → X^{an},

denote the universal covering space of Y^{an} (resp. X^{an}) corresponding to the basepoint o. The group Γ acts on Ỹ transitively and without fixed points, and the map γ ↦ γ(1) identifies the quotient Ỹ /Γ with Y^{an}.

Recall that a closed, C-valued smooth 1-form (resp. a meromorphic 1-form of the second kind) η on X^{an} admits a smooth (resp. meromorphic) primitive F_{η} : X̃ → C, defined by the rule

F_{η}(γ) := ∫_{0}^{1} γ^{∗}η.

Definition 2.2.2. The basic iterated integral attached to an ordered n-tuple (ω_{1}, …, ω_{n}) of smooth 1-forms on Y^{an} is the function P(Y; o) → C, denoted ∫ ω_{1} · ω_{2} · . . . · ω_{n}, defined by

γ ↦ ∫_{γ} ω_{1} · ω_{2} · . . . · ω_{n} := ∫_{Δ}(γ^{∗}ω_{1})(t_{1})(γ^{∗}ω_{2})(t_{2}) ⋯ (γ^{∗}ω_{n})(t_{n}),

where Δ is the simplex in [0, 1]^{n} defined by 0 ≤ t_{n} ≤ t_{n−1} ≤ ⋯ ≤ t_{1} ≤ 1. The integer n is called the length of this basic iterated integral.

Example 2.2.3. When n = 2, the basic iterated integral attached to ω and η can be computed by the formula

∫_{γ} ω · η = ∫_{γ} ωF_{η} = ∫_{0}^{1} γ^{∗}(ωF_{η}).

In the expression in the middle, we abusively use the same notation ω for the differential π^{∗}ω on Ỹ. The 1-form ωF_{η} is to be integrated along a lift of γ to Ỹ, which is unique once a lift of o to Ỹ is specified.

Definition 2.2.4. An iterated integral is a linear combination of basic iterated integrals, viewed as a function on P(Y; o). Its length is defined to be the maximum of the lengths of its constituent basic iterated integrals. It is said to be homotopy invariant if its value on any path γ depends only on the homotopy class of γ.

A homotopy-invariant iterated integral defines a C-valued function on Γ, and by extending linearly induces a homomorphism of abelian groups Z[Γ] → C. Observe that a homotopy invariant iterated integral of length ≤ n vanishes on the (n + 1)^{st} power I^{n+1} of the augmentation ideal in Z[Γ], and hence gives rise to a well-defined element of Hom(I/I^{n+1}, C). The natural map

(2.2.1)  {homotopy invariant iterated integrals of length ≤ n} ⟶ Hom(I/I^{n+1}, C)

is an isomorphism; see, for example, [H2].

We will be interested in numerically evaluating certain homotopy invariant iterated integrals on Y of length ≤ 2. Suppose ω and η are two differentials of the second kind on X, regular on Y, representing cohomology classes ω, η ∈ H_{dR}^{1}(X/Q) in the manner of §2.1. The basic iterated integral ∫ ω · η of length 2 is not generally homotopy invariant. But when either ω or η is holomorphic on X — i.e., has no pole at ∞ — a suitable modification of ∫ ω · η will be homotopy invariant, as we now explain.

Recall that a differential on a Riemann surface is said to have a logarithmic pole at a point if its expansion in terms of a local parameter q at this point is of the form ∑_{n=0}^{∞} a_{n}q^{n} \frac{dq}{q}. When ω is holomorphic at ∞, we let α_{ω,η} be a meromorphic 1-form on X that is regular on Y and is such that the induced differential ωF_{η} − α_{ω,η} on X̃ has at worst a logarithmic pole at (any point lying over) ∞. This condition is well-posed because the principal part of ωF_{η} at x̃ ∈ X̃ depends only on the image x of x̃; see [DRS, §2]. The form α_{ω,η} exists — and in fact can even be taken to be algebraic and defined over Q — by Riemann-Roch. If ω is not holomorphic at ∞ but η is, then we define α_{ω,η} := −α_{η,ω}.

Lemma 2.2.5. Let ω and η be as above, and assume that either ω or η is holomorphic at ∞. Then the iterated integral J_{ω,η} := ∫ ω · η − α_{ω,η} is homotopy-invariant.

Proof. The homotopy invariance of J_{ω,η} follows from the fact that J_{ω,η}(γ) = ∫_{γ} ωF_{η} − α_{ω,η}, and the 1-form on X̃ in the integrand is holomorphic when restricted to Ỹ.

□

Remark 2.2.6. Note that if ω and η are both holomorphic at ∞, then we can take α_{ω,η} = 0.

Now consider an integral Hodge class ξ ∈ H^{1}(X^{an}, Z) ⊗ H^{1}(X^{an}, Z). The Hodge condition implies that we can choose a basis {ω_{i}} such that when we write ξ = ∑ c_{i,j}ω_{i} ⊗ω_{j}, then either ω_{i} or ω_{j} is holomorphic at ∞ whenever c_{i,j} ≠ 0. By the previous lemma, the iterated integral J_{ξ} = ∑ J_{ω_{i},ω_{j}} is homotopy invariant

Lemma 2.2.7. Suppose that ξ is an integral Hodge class on X × X as above. Using (2.2.1), identify J_{ξ} with a homomorphism of abelian groups I/I^{3} → C. Then the restriction of J_{ξ} to I^{2}/I^{3} is Z-valued and agrees with ξ viewed as an element of

H^{1}(X^{an}, ℤ) ⊗ H^{1}(X^{an}, ℤ) ≅ (H_{1}(X^{an}, ℤ) ⊗ H_{1}(X^{an}, ℤ))^{∨}
= (I/I^{2} ⊗ I/I^{2})^{∨} = (I^{2}/I^{3})^{∨}.

(Here A^{∨} denotes Hom(A, ℤ), for any abelian group A.)

Proof. See the discussion at the beginning of §2 of [DRS], and loc. cit., Lemma 1.1(2). □

By Lemma 2.2.7, the map J_{ξ} induces a homomorphism

J_{ξ} : H_{1}(X^{an}, Z) = I/I^{2} → C/Z.

The following observation, which is extended in Theorem 2.3.1 below to the entire Jacobian of X, is key in our approach to calculating Chow-Heegner points. Fix any holomorphic 1-form ρ ∈ H^{1,0}(X_{C}) ⊂ H^{1}(X^{an}, C) corresponding to an elliptic curve factor E of the Jacobian of X, and denote by Λ the period lattice

Λ := {∫_{γ} ρ, γ ∈ H_{1}(X^{an}, ℤ)} ⊂ C

attached to ρ. The class γ_{ρ} ∈ H_{1}(X^{an}, C) that is Poincaré dual to ρ actually belongs to H_{1}(X^{an}, Z) ⊗ Λ. Consequently J_{ξ}(γ_{ρ}) can be viewed as a well-defined element of C/Λ, and hence of E(ℂ).

2.3. Chow-Heegner points. Let X_{1}, X_{2} denote copies of X, and X_{12} the diagonal copy of X in X_{1} × X_{2}. We fix a choice of base point ∞ ∈ X and use it to identify X_{1} and X_{2} with the curves X_{1} × {∞} and {∞} × X_{2} respectively in X_{1} × X_{2}. To a codimension-one algebraic cycle Z ⊂ X × X = X_{1} × X_{2} (defined over Q) we associate the divisor class

D_{Z} = (Z ∩ X_{12}) − (Z ∩ X_{1}) − (Z ∩ X_{2}).

Define a corresponding degree-zero divisor class

P_{Z} = D_{Z} − deg(D_{Z})o ∈ Pic^{0}(X).

(Recall that o ∈ X(Q) is a fixed base point which is different from the previously chosen base point ∞.)

We now state the iterated integral formula from [DRS] for the image of P_{Z} under the Abel-Jacobi map

AJ_{X} : Pic^{0}(X) → Ω^{1}(X^{an})^{∨}/H_{1}(X^{an}, Z).

Let ϵ_{o} be the projector on Pic(X × X) defined by

ϵ_{o}(Z) = Z − i_{1∗}π_{1∗} − i_{2∗}π_{2∗}

where π_{1}, π_{2} : X × X ⇉ X are the projections and i_{1}, i_{2} : X ⇉ X × X are the inclusions of “vertical and horizontal” copies of X over the basepoint o.

Let

cl(ϵ_{o}−) : Pic(X × X) → H^{1}(X^{an}, Z) ⊗ H^{1}(X^{an}, Z)

denote the composition of the cycle class map and the projector ϵ_{o}. (The effect of ϵ_{0} is to annihilate the H^{2} ⊗ H^{0} and H^{0} ⊗ H^{2} factors in the Künneth decomposition of the second Betti cohomology group of X × X.) Suppose cl(ϵ_{o}Z) is represented by ∑_{i,j} c_{i,j}ω_{i} ⊗ ω_{j}, where ω_{i} or ω_{j} is holomorphic at ∞ whenever c_{i,j} ≠ 0.

Theorem 2.3.1 ([DRS], Corollary 3.6). The image AJ_{X}(P_{Z}) ∈ Ω^{1}(X^{an})^{∨}/H_{1}(X^{an}, Z) is represented by the linear functional that maps ρ ∈ Ω^{1}(X^{an}) to

∑_{i,j} c_{i,j} ∫_{γ_{ρ}}(ω_{i} · ω_{j} − α_{ω_{i},ω_{j}}) + deg(D_{Z}) ∫_{o}^{∞} ρ ∈ C,

where γ_{ρ} ∈ H_{1}(X^{an}, C) is Poincaré dual to ρ ∈ H^{1,0}(X^{an}) ⊂ H_{dR}^{1}(X^{an}, C). □

3. Chow-Heegner points on modular curves

We now specialize the discussion of the preceding section to the case of classical modular curves X. We shall define certain rational points on an arbitrary elliptic curve E_{/Q} called Chow-Heegner points, such that the corresponding points of E(C) ≅ C/Λ_{E} can be computed using iterated path integrals via Theorem 2.3.1.

3.1. Definitions. Let N > 1 be an integer and X = X_{0}(N) denote the canonical model over Q of the classical modular curve of level N; write J_{0}(N) for the Jacobian of X_{0}(N). With this choice of X we place ourselves in the setup of §1, taking the ground field F to be Q and the point ∞ ∈ X(Q) to be the usual cusp at infinity. Thus Y := X_{0}(N) − {∞}. (Note that Y differs from the usual open modular curve Y_{0}(N).) For the moment we do not specify the basepoint o ∈ Y^{an} used for topological constructions.

We write S_{2}(Γ_{0}(N)) for the space of cuspidal weight 2 modular forms for Γ_{0}(N). This space is canonically isomorphic to the vector space Ω^{1}(X^{an}) of holomorphic 1-forms on X, via the map which associates to a modular form f : ℌ^{∗} → C the 1-form ω_{f} = 2πif(z) d z on X. We recall that if f has Fourier expansion ∑_{n≥1} a_{n}q^{n} then a Laurent expansion for ω_{f} near ∞ is given by ∑_{n≥1} a_{n}q^{n} \frac{d q}{q}.

Let E be an elliptic curve over Q of conductor N_{E} dividing N. Its isogeny class corresponds to a newform f ∈ S_{2}(Γ_{0}(N_{E})) with rational Fourier coefficients, which gives rise to a modular parametrization π_{E} : J_{0}(N_{E}) → E, a morphism of abelian varieties defined over Q.

For the ease of exposition, we will assume throughout that ker π_{E} is connected. In this case, assuming the Manin constant c = 1, the Néron lattice of E coincides with the period lattice Λ_{f} of the differential ω_{f} = 2πif(z)dz ∈ Ω^{1}(X_{0}(N_{E})^{an}) corresponding to f.

The map π_{E} can be computed on complex points explicitly, using the Abel-Jacobi isomorphism

AJ_{X} : J_{0}(N_{E})(C) ≅ Ω^{1}(X_{0}(N_{E})^{an})^{∨}/H_{1}(X_{0}(N_{E})^{an}, Z),

the Weierstrass uniformization W : C/Λ_{f} ≅ E(C), and the analytic parametrization

π_{E}^{an} : Ω^{1}(X_{0}(N_{E})^{an})^{∨}/H_{1}(X_{0}(N_{E})^{an}, Z) → C/Λ_{f}.

The map π_{E}^{an} sends the coset of a functional on Ω^{1}(X_{0}(N_{E})^{an}) to the evaluation of that functional at ω_{f}. Thus for P_{C} ∈ J_{0}(N_{E})(C) we have

(3.1.1)  π_{E,C}(P_{C}) = W (π_{E}^{an}(AJ_{X_{0}(N_{E})}(P_{C}))) = W (AJ_{X_{0}(N_{E})}(P_{C})(ω_{f})).

For each divisor d of N/N_{E} there is a degeneracy map π_{d} : X_{0}(N) → X_{0}(N_{E}) that induces by the Albanese universal property a morphism of abelian varieties π_{d} : J_{0}(N) → J_{0}(N_{E}), which we denote with the same symbol. Put π_{E}^{d} = π_{E} · π_{d} : J_{0}(N) → E for the composition of π_{d} with π_{E}. When d = 1, the map π_{1} is the projection arising from the natural inclusion Γ_{0}(N) ⊆ Γ_{0}(N_{E}) and we shall simply write π_{E}^{1} = π_{E}; it will be clear from the context whether the source of this map is J_{0}(N) or J_{0}(N_{E}).

Let T = Q[…, T_{n}, …] be the Hecke algebra of level N, and let T_{0} denote the subalgebra generated by Hecke operators prime to the level N. Then

T_{0} ≃ ∏_{h} K_{h} ⊆ T ≃ ∏_{h} L_{h}

where h runs over Galois conjugacy classes of newforms of all levels M dividing N,

K_{h} = Q({a_{n}(h)}_{n≥1})

is the number field generated by the Hecke eigenvalues of h, and, if N_{h} | N stands for the level of h, the ring L_{h} is a commutative Artinian K_{h}-algebra of dimension [L_{h} : K_{h}] = σ(N/N_{h}), the number of factors of N/N_{h} (cf. [Ka], [RS, Ch. 14]). Let End(Jac(X)) denote the ring of endomorphisms of Jac(X) which are defined over ℚ. The endomorphism algebra End^{0}(Jac(X)) := Q ⊗ End(Jac(X)) contains the Hecke algebra T_{0} as its center and T as a maximal commutative subalgebra (and End^{0}(Jac(X)) ≃ T ≃ T_{0} when N is prime), as

End^{0}(Jac(X)) = ⟨T, {δ_{d}}_{d|σ(N/N_{h})}⟩ ≃ ∏_{h} M_{σ(N/N_{h})}(K_{h}),

where δ_{d} are the degeneracy operators as defined e.g. in [Ka].

Definition 3.1.1. For a divisor M of N and a newform g ∈ S_{2}(Γ_{0}(M)), denote by

T_{g} ∈ T_{0} ≅ ∏_{h} K_{h}

the idempotent with 1 in the K_{g} component and 0 elsewhere.

We shall also view T_{g} as an idempotent of T and of End^{0}(Jac(X)) by the natural inclusion, so that T[g] := T_{g} · T = L_{g} and End^{0}(Jac(X))[g] := T_{g} · End^{0}(Jac(X)) = M_{σ(N/M)}(K_{g}).

There are natural isomorphisms

(3.1.2) End(Jac(X)) ⊗ Q ≃ Corr(X) := Pic(X × X) ⊗ Q/(π_{1}^{∗}Pic(X) ⊗ Q ⊕ π_{2}^{∗}Pic(X) ⊗ Q).

Under these identifications, an endomorphism T ∈ End^{0}(Jac(X)) is associated to a divisor class in Pic(X × X) ⊗ Q, which we also denote by the same symbol, and is well-defined only up to horizontal and vertical divisors.

In §2.3 a point P_{T} ∈ Pic^{0}(X) = J_{0}(N)(Q) was attached to a divisor T on X × X; this point only depends on the class of T in Pic(X × X) ⊗ Q/(π_{1}^{∗}Pic(X) ⊗ Q ⊕ π_{2}^{∗}Pic(X) ⊗ Q). This construction gives rise to a map

Pic(X × X) ⊗ Q → J_{0}(N)(Q) ⊗ Q
T ↦ P_{T} .

Note that both these divisor classes and the projector ϵ_{o} depend on the basepoint o, so we must now specify a particular choice of o. In the following definition, we fix the choice of basepoint o to be the cusp 0 ∈ X, which is distinct from ∞ because N > 1.

Definition 3.1.2. For an operator T ∈ End^{0}(Jac(X))[g] and a divisor d of N/N_{E}, define the Chow-Heegner point

P_{T,f^{(d)}} = π_{E}^{d}(P_{T}) ∈ E(Q) ⊗ Q

When we take d = 1 we shall just write P_{T,f} for P_{T,f^{(d)}}; in addition, the Chow-Heegner point associated with T = T_{g} and d = 1 shall be denoted P_{g,f}.

For any positive integer n, define T_{g,n} := T_{g} · T_{n} ∈ T as the product of T_{g} and T_{n}, and define

P_{g,f,n} := P_{T_{g,n},f}.

Note that P_{g,f} = P_{g,f,1} and that P_{g,f,n} is a multiplie of P_{g,f} if the eigenvalue a_{n}(g) of T_{n} acting on the g-isotypic component is rational. Thus, the points P_{g,f,n} for n > 1 are only of interest when g has irrational Fourier coefficients or is an oldform. For a general discussion of the notion of Chow-Heegner point, of which the construction above is a special case, see the introduction of [BDP2] and [DRS].

Remark 3.1.3. To avoid the tensor products with Q in the definition of the above Chow-Heegner points P_{T,f^{(d)}}, one must work with an actual algebraic cycle on X × X, as opposed to T which is merely a Q-linear combination of such cycles. Define the “denominator” d_{T} of T ∈ T to be the smallest positive integer such that d_{T} T lies in the integral Hecke algebra T_{Z} := Z[…, T_{n}, …]. Then d_{T} P_{T,f^{(d)}} belongs to E(Q); see also §4.7.

However, we regard d_{g,n}P_{g,f,n} as well-defined only modulo the torsion subgroup E(Q)^{tor}. One reason for this is that to obtain a computable formula for P_{g,f,n} one can take the basepoint o to be any cusp other than ∞. There being no distinguished choice, one is left with an ambiguity valued in the image in E(Q) of the (torsion) cuspidal subgroup of J_{0}(N); see (3.2.2) below and the subsequent remarks.

Theorem 3.1.4 ([DRS], Corollary 1.4). Assume that the local signs ε_{p}(g, g, f) of Garrett’s triple product L-function L(g, g, f, s) are +1 at the primes p | N. Then the module of points

\underline{P}_{g,f} := ⟨P_{T,f^{(d)}} : T ∈ End^{0}(Jac(X))[g], d | \frac{N}{N_{E}} ⟩ ⊆ E(Q) ⊗ Q

is nonzero (equivalently, there is a point ∑_{d|N/N_{E}} n_{d}π_{E}^{d}(d_{T} P_{T}) ∈ E(Q) which is non-torsion, for some T and integers n_{d}) if and only if the following conditions hold:

i. L(f, 1) = 0, ii. L′(f, 1) ≠ 0, and iii. L(f ⊗ Sym^{2}(^{σ}g), 2) ≠ 0 for all σ ∈ Gal (Q̄/Q).

3.2. Algorithms. The aim of this section is to explain how to explicitly compute the slightly smaller submodule

\underline{P}_{T[g],f} := \{P_{T,f} : T ∈ T[g]\} = ⟨P_{g,f,n} : n ≥ 1⟩ ⊆ \underline{P}_{g,f}

of the Mordell-Weil group E(Q) ⊗ Q of E. Although the methods generalize to the computation of the full module of points \underline{P}_{g,f}, the computation of \underline{P}_{T[g],f} is simpler, and often sufficient for the purpose of finding a non-torsion point when one exists by Theorem 3.1.4. Indeed, the authors have not come across an example where \underline{P}_{T[g],f} is torsion but the a priori larger \underline{P}_{g,f} is non-torsion. The interested reader may consult [Ka] for details relevant to the eventual adaptations of our methods to calculating \underline{P}_{g,f} in its entirety.

Theorem 2.3.1 gives rise to an explicit formula for a Chow-Heegner point P_{g,f,n} in terms of iterated integrals. This formula is stated in terms of the components of the cohomology class cl(ϵ_{o}T_{g,n}) ∈ H_{dR}^{1}(X/Q)^{⊗2} expressed as a sum of pure tensors. In this section we find an explicit formula for cl(ϵ_{o}T_{g,n}).

The action of the Hecke algebra T_{0} on modular forms extends to an action on the de Rham cohomology of X. Under this action, we have

H_{dR}^{1}(X/Q) ≅ H_{dR}^{1}(X/Q)[g_{1}] ⊕ ⋯ ⊕ H_{dR}^{1}(X/Q)[g_{n}],

indexed by Galois conjugacy classes of newforms of all levels M dividing N. Let g ∈ S_{2}(Γ_{0}(M)) be a newform and suppose {ω_{g,1}, …, ω_{g,2k}} is a collection of differentials of the second kind on X representing a basis for H_{dR}^{1}(X/Q)[g]. Write T_{n}ω_{g,i} = ∑_{j} a_{ij}^{n}ω_{g,j}, and denote by A_{n} and B the matrices (a_{ij}^{n})_{1≤i,j≤2k} and (⟨ω_{g,i}, ω_{g,j}⟩)_{1≤i,j≤2k}, respectively, where ⟨ , ⟩ denotes the Poincaré pairing. Then we have the following:

Lemma 3.2.1. cl(ϵ_{o}T_{g,n}) = ∑_{i,j} c_{ij}^{n}ω_{g,i} ⊗ ω_{g,j}, where (c_{ij}^{n})_{1≤i,j≤2k} = −B^{−1}A_{n}.

Proof. The projector ϵ_{o} acts on H_{dR}^{2}(X × X) by annihilating the H_{dR}^{0}(X) ⊗ H_{dR}^{2}(X) and H_{dR}^{2}(X) ⊗ H_{dR}^{0}(X) components of the Künneth decomposition, so we have

cl(ϵ_{o}T_{g,n}) ∈ H_{dR}^{1}(X) ⊗ H_{dR}^{1}(X).

Note from the definition that T_{g,n} acts on H_{dR}^{1}(X)[h] as T_{n} if h = g and 0 otherwise, so cl(ϵ_{o}T_{g,n}) is equal to the image of T_{n} under the identification:

End(H_{dR}^{1}(X)[g]) ≃ H_{dR}^{1}(X)[g]^{∨} ⊗ H_{dR}^{1}(X)[g] ≃ H_{dR}^{1}(X)[g] ⊗ H_{dR}^{1}(X)[g]

The first map is the canonical isomorphism of finite dimensional vector spaces, and the second is induced from the inverse of the identification H_{dR}^{1}(X)[g] ≃ H_{dR}^{1}(X)[g]^{∨} via the map v ↦ (w ↦ ⟨v, w⟩). The remainder of the proof is a straightforward exercise in linear algebra, and is left to the reader. □

Combining the previous results, we obtain the following formula for P_{g,f,n}. Let γ_{f} be the Poincaré dual of ω_{f} and let ω_{g,1}, …, ω_{g,2k} be differentials of the second kind that give rise to a symplectic basis for the g-isotypic Q-subspace H_{dR}^{1}(X/Q)[g]. That is, for 1 ≤ i ≤ k we have ⟨ω_{g,i}, ω_{g,i+k}⟩ = 1 and ⟨ω_{g,i}, ω_{g,j}⟩ = 0 for j ≠ i + k. Assume moreover that this basis is adapted to the Hodge filtration, in the sense that the 1-forms ω_{g,1}, …, ω_{g,k} are regular at ∞. Since H^{0}(X, Ω^{1}(X/Q))[g] ⊂ H_{dR}^{1}(X/Q)[g] is a maximal isotropic subspace with respect to the Poincaré pairing, we can simply take any basis ω_{g,1}, …, ω_{g,k} for H^{0}(X, Ω^{1}(X/Q))[g] = S_{2}(Γ_{0}(N))[g] and extend it to a symplectic basis of the sort desired. Write

(3.2.1)  cl(ϵ_{o}T_{g,n}) = ∑_{i,j}^{2k} c_{ij}^{n}ω_{g,i} ⊗ ω_{g,j}

where the coefficients can be computed as in Lemma 3.2.1. Since we chose the basis to be symplectic, the matrix B is the standard symplectic matrix. Additionally, because T_{n} preserves the holomorphic subspace of H_{dR}^{1}(X/Q), then the computation of −B^{−1}A_{n} shows that c_{ij}^{n} = 0 when k + 1 ≤ i, j ≤ 2k. Thus, the Hodge class in (3.2.1) is of the form discussed after Remark 2.2.6.

Combining (3.2.1) with (3.1.1) and Theorem 2.3.1, we obtain the following formula for the point P_{g,f,n}:

(3.2.2)  P_{g,f,n} = W (∑_{i,j}^{2k} c_{ij}^{n} (∫_{γ_{f}} ω_{g,i} · ω_{g,j} − α_{ω_{g,i},ω_{g,j}})) .

Note that (3.2.2) omits the term deg(D_{T_{g,n}}) ∫_{o}^{∞} ω_{f} from the formula in Theorem 2.3.1. This is justified by our choice of basepoint o, for it is well-known (cf. e.g. [Man]) that the difference [∞] − [0] is a torsion point of J_{0}(N)(Q). Since P_{g,f,n} is defined as an element of E(Q) ⊗ Q, torsion can be disregarded.

We emphasize that by Lemmas 2.2.5 and 2.2.7, the right-hand side of (3.2.2) depends only on the homology class

γ_{f} ∈ H_{1}(Y^{an}, Z) = H_{1}(X^{an}, Z)

Poincaré dual to ω_{f}. It can therefore be evaluated by lifting γ_{f} arbitrarily to an element γ̃_{f} ∈ π_{1}(Y^{an}; o) and evaluating the length ≤ 2 homotopy invariant iterated integral (3.2.2) on any loop in the homotopy class γ̃_{f}.

Remark 3.2.2. When n = 1, formula (3.2.2) greatly simplifies. In the notation of Lemma 3.2.1, A_{1} is the identity matrix and B is the standard symplectic matrix by our choice of basis, and so if we write η_{g,i} for ω_{g,i+k}, then

P_{g,f} = W (∑_{i=1}^{k} (∫_{γ_{f}} ω_{g,i} · η_{g,i} − η_{g,i} · ω_{g,i} − 2α_{ω_{g,i},η_{g,i}}))

Although P_{g,f,n} is defined relative to the choice of base point o as the cusp 0 on Y, it will be convenient to compute it relative to a different choice of basepoint.

Lemma 3.2.3. The right hand side of (3.2.2) is independent of the choice of o ∈ Y (Q).

Proof. Changing the basepoint from o to o′ amounts to conjugating the representative path γ_{f} for the homology class Poincaré dual to ω_{f} by a path β from o to o′. This manifestly does not affect the value of the integral of the meromorphic 1-form α_{ω_{g,i},ω_{g,j}}. Thus the issue is whether we have an identity

(3.2.3)  ∫_{γ_{f}} ω_{g,i} · ω_{g,j} =^{?} ∫_{βγ_{f} β^{−1}} ω_{g,i} · ω_{g,j}.

But by [H1, Exer. 8], for any 1-forms ω, η, loop γ, and path β, we have

(3.2.4)  ∫_{βγβ^{−1}} ω · η = ∫_{γ} ω · η + | \begin{matrix}∫_{γ} ω & ∫_{γ} η \\ ∫_{β} ω & ∫_{β} η \end{matrix} | .

In our situation, the determinants expressing the difference between the two sides of (3.2.3) vanish. Indeed, ∫_{γ_{f}} ω_{g,i} = ⟨ω_{f}, ω_{g,i}⟩ = 0, since the decomposition into isotypic components for the action of the Hecke algebra is orthogonal with respect to the Poincaré pairing. □

4. Details of the algorithm

We now turn to the question of numerically evaluating formula (3.2.2) for a Chow-Heegner point P_{g,f,n} ∈ E(Q) ⊗ Q for an elliptic curve E = E_{f}. We retain all the notation from §§2–3. The following ingredients occur in (3.2.2):

1. The Poincaré dual γ_{f} ∈ H_{1}(X, C) of ω_{f} ∈ H_{dR}^{1}(X^{an}, C).

2. A collection of rational differentials of the second kind ω_{g,1}, …, ω_{g,2k} on X, regular away from ∞, whose images in H_{dR}^{1}(X/Q) are a symplectic basis for the g-isotypic component H_{dR}^{1}(X/Q)[g]. This basis should be adapated to the Hodge filtration on H_{dR}^{1}(X/Q)[g] in the sense that ω_{g,1}, …, ω_{g,k} are holomorphic everywhere on X, including the point ∞.

3. The coefficients c_{ij}^{n} arising from the action of T_{n} on H_{dR}^{1}(X) with respect to the basis ω_{g,1}, …, ω_{g,2k}.

4. Meromorphic differentials α_{ω_{g,i},ω_{g,j}} on X, regular on Y, such that ω_{g,i}F_{η_{g,j}} − α_{ω_{g,i},η_{g,j}}

has at worst a logarithmic pole at (any point lying over) ∞ for 1 ≤ i ≤ k. These data must be “known” in a sufficiently concrete form to evaluate the iterated integrals occuring in (3.2.2). It is also desirable to know

5. the denominator d_{g,n} of the projector onto the g-isotypic component of the cohomology of X.

This last item will allow for the computation of a point in E(Q), as opposed to one in E(Q) ⊗ Q. This section is devoted to methods of computing these five ingredients.

4.1. Evaluating iterated integrals. Let J = ∫ ω·η−α_{ω,η} be a homotopy-invariant iterated integral of length ≤ 2 on Y, expressed in terms of differentials of the second kind on X, regular on Y. We seek to compute the right-hand side of formula (3.2.2), which is a Qlinear combination of J(γ) for various choices of J and (the homotopy class of) a path γ ∈ π_{1}(Y; o). As remarked earlier, (3.2.2) actually depends only on the homology class γ_{0} of γ. This homology class belongs to H_{1}(Y^{an}, Z) = H_{1}(X^{an}, Z), which is the abelianization of the quotient π_{1}(X^{an}, o) = Γ̄_{0}(N) of Γ_{0}(N) by the smallest normal subgroup containing the elliptic and parabolic elements.

To evaluate J(γ_{0}) for γ_{0} ∈ H_{1}(Y^{an}, Z), choose the basepoint o away from the set S of elliptic points on Y_{0}(N)^{an} ⊂ Y^{an} and lift γ_{0} arbitrarily to a path γ̃ ∈ π_{1}(Y_{0}(N)^{an} \ S, o). For each elliptic point x ∈ S, let e_{x} = |Stab_{Γ_{0}(N)}(x)/{±1}| denote the index of x (which is either 2 or 3) and γ_{x} be a sufficiently small counterclockwise loop around x. Writing H for the normal subgroup of π_{1}(Y_{0}(N)^{an} \ S, o) generated by {γ_{x}^{e_{x}}, x ∈ S}, there is a natural isomorphism Γ_{0}(N) ≃ π_{1}(Y_{0}(N)^{an} − S; o)/H.

We may regard then γ̃ as an element of Γ_{0}(N); this causes no ambiguity because H lies in the kernel of the natural projection H_{1}(Y_{0}(N)^{an} − S, Z) → H_{1}(Y_{0}(N)^{an}, Z) → H_{1}(Y^{an}, Z). The path γ̃ can then also be viewed as a path in ℌ from τ_{0} to γτ_{0}, where γ ∈ Γ_{0}(N) is a lift of γ_{0}.

Lemma 4.1.1. Suppose γ_{0} is Poincaré-dual to ρ. As an element of C/Λ_{ρ}, we have

J(γ_{0}) = ∫_{τ_{0}}^{γτ_{0}} ωF_{η} − α_{ω,η}

where we conflate 1-forms on X with their pullbacks to ℌ^{∗} = ℌ ∪ {∞}. Moreover, F_{η} has Laurent expansion about ∞ ∈ 𝔥^{∗} given by formally integrating the Laurent expansion of η about the cusp ∞ ∈ X.

Proof. This follows from the preceding discussion, using the definition of iterated integrals and the homotopy invariance of J. □

Given any differential form λ of the second kind on X, and any γ ∈ Γ_{0}(N), let

I(λ; γ) := ∫_{τ_{0}}^{γτ_{0}} λ.

(As above, in the righthand side of this expression λ is conflated with its pullback to ℌ^{∗}.) By the residue formula, this expression is independent of the choice of path on the upper half-plane ℌ from τ_{0} to γτ_{0}. The Γ_{0}(N)-invariance of λ also shows that this expression is independent of the choice of base point τ_{0} ∈ ℌ, which justifies suppressing τ_{0} from the notation.

If λ instead denotes a differential of the second kind on X̃ then the integral above still makes sense but depends on both the basepoint o and the chosen lift of o to τ_{0} ∈ ℌ. We will primarily be interested in evaluating such integrals in the context of (3.2.2), for which the choice of basepoint is ultimately irrelevant. (This is because the Poincaré dual of the homology class of γ, is orthogonal to the 1-forms in the iterated integral giving rise to the path integral we seek to evaluate; cf. Lemma 3.2.3.) However, as we are about to see, for the purposes of algorithmic efficiency it is necessary to break up the path of integration into pieces that can be computed relatively quickly. The integrals over these pieces may no longer be basepoint-independent: when we express γ as a product of computationally-amenable elements γ^{(j)} ∈ Γ_{0}(N), the corresponding homology classes may no longer lie in (the Poincaré dual of) the orthogonal complement of H_{dR}^{1}(X/Q)[g]. Thus for a general meromorphic 1-form λ on X̃ and a general γ ∈ Γ_{0}(N), we adopt the notation

I_{τ_{0}}(λ; γ) = ∫_{τ_{0}}^{γτ_{0}} λ

to emphase the dependence on the choice of basepoint.

By meromorphicity, for λ as above (defined on either X or X̃) the integral I(λ; γ) or I_{τ_{0}}(λ; γ) can be computed by integrating term-wise a Laurent expansion for λ using the fundamental theorem of calculus. Thus, in practice, one computes the Laurent expansion for the primitive F_{λ} about ∞ ∈ X (or a choice of ˜∞ ∈ X̃ lying over ∞), regarded as function given by a convergent power series in q = e^{2πiτ} on 𝔥, and evaluates it at τ_{0} and τ′_{0} = γτ_{0}. The larger the imaginary parts of τ_{0} and τ′_{0} are, the faster this series converges and the fewer coefficients of the Laurent series of λ are necessary to approximate I(λ; γ) or I_{τ_{0}}(λ; γ) to a give degree of accuracy. Writing γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ), it is well-known that the best compromise between Im(τ_{0}) and Im(τ′_{0}) is achieved when we choose τ_{0} = −\frac{d}{c} + \frac{1}{|c|}i (cf., for example, [Cr, p. 35]). This optimal basepoint for γ will be denoted τ_{γ}^{∗}.

With this remark in mind, we take the following approach to computing J(γ_{0}) as in the lemma above. First compute Laurent expansions for the differentials ω, η, α_{ω,η}. Then find a “good” expression for the homology class γ_{0} ∈ H_{1}(X^{an}, C), writing it as a C-linear combination of classes γ_{0}^{(j)} ∈ H_{1}(X^{an}, Z) that lift to elements γ^{(j)} ∈ Γ_{0}(N) with small lower-left entries cN. Finally, calculate approximations to the integrals I_{τ_{0}}(ωF_{η}; γ^{(j)}) and I(α_{ω,η}; γ^{(j)}). The appropriate linear combination of these integrals is an (approximate) representative for the coset J(γ_{0}) ∈ C/Λ_{ρ}.)

To calculate I(α_{ω,η}; γ^{(j)}), one is free to change the basepoint from τ_{0} to the optimal basepoint τ_{j}^{∗} := τ_{γ^{(j)}}^{∗} for γ^{(j)}, since α_{ω,η} is defined on X and not only on X̃. The same is not true

for ωF_{η}. To evaluate I_{τ_{0}}(ωF_{η}; γ^{(j)}) we appeal to the following lemma.

Lemma 4.1.2. I_{τ_{0}}(ωF_{η}; γ^{(j)}) = I_{τ_{j}^{∗}}(ωF_{η}; γ^{(j)}) − I(η; γ^{(j)}) ∫_{τ_{0}}^{τ_{j}^{∗}} ω.

Observe that every term on the righthand side can be computed using the fundamental theorem of calculus, evaluating power series only at the points τ_{0} and τ_{j}^{∗}. In particular, taking τ_{0} = i/N, each such evaluation converges at least as fast as an evaluation at τ_{j}^{∗}, so this formula for the integral is “optimally efficient”.

Proof of Lemma 4.1.2. Since λ = ωF_{η} is a holomorphic 1-form on ℌ, its integral along a closed contour vanishes. Thus

I_{τ_{0}}(λ; γ^{(j)}) = I_{τ_{j}^{∗}}(λ; γ^{(j)}) + ∫_{τ_{0}}^{τ_{j}^{∗}} λ − ∫_{γ^{(j)}τ_{0}}^{γ^{(j)}τ_{j}^{∗}} λ.

To evaluate the second term on the righthand side, we observe that ω comes from a 1-form on X, so it is Γ_{0}(N)-invariant; it thus pulls back to itself along the fractional linear transformation defined by γ^{(j)}. On the other hand,

I(η; γ^{(j)}) = ∫_{γ^{(j)}} η = F_{η}(γ^{(j)}τ) − F_{η}(τ), for all τ ∈ ℌ.

Hence (γ^{(j)})^{∗}F_{η} = F_{η} + I(η; γ^{(j)}). So (γ^{(j)})^{∗}λ = λ + I(η; γ^{(j)})ω, and we find

∫_{γ^{(j)}τ_{0}}^{γ^{(j)}τ_{j}^{∗}} λ = ∫_{τ_{0}}^{τ_{j}^{∗}}(γ^{(j)})^{∗}λ = ∫_{τ_{0}}^{τ_{j}^{∗}} λ + I(η; γ^{(j)}) ∫_{τ_{0}}^{τ_{j}^{∗}} ω,

which yields the lemma. □

Remark 4.1.3. We warn the reader that possibly I_{τ_{0}}(ωF_{η}; γ^{(j)}) ≠ ∫_{γ^{(j)}} ω · η (regarding γ^{(j)} as an element of π_{1}(Y^{an}; o)). Indeed, the iterated integral attached to ω · η need not even be homotopy invariant (!) so ∫_{γ^{(j)}} ω·η is ill-defined. In particular, one cannot relate I_{τ_{0}}(ωF_{η}; γ^{(j)}) to I_{τ_{j}^{∗}}(ωF_{η}; γ^{(j)}) using the change-of-basepoint formula (3.2.4) for iterated integrals.

To efficiently evaluate the integrals in (3.2.2) using the method just explained, it is therefore necessary to know:

a. Laurent expansions about ∞ for a symplectic basis ω_{g,1}, …, ω_{g,2k} of H_{dR}^{1}(X/Q)[g]

and the forms α_{ω_{g,i},ω_{g,j}}; and, b. the homology class γ_{f} as a C-linear combinationation of class γ_{0}^{(j)} whose lifts to Γ_{0}(N)

have small lower-left entries cN. In the rest of this section we turn to the task of computing these data.

4.2. Calculating a symplectic basis for H_{dR}^{1}(X/Q)[g]. The calculation of a basis for the de Rham cohomology can be carried out by first writing down a modular function u — that is, a rational function on X = X_{0}(N) — which is regular away from ∞. Such a function exists by Riemann-Roch and a q-expansion for one such function can sometimes be computed explicitly using the Dedekind eta-function, as explained in the next subsection.

Using a modular symbol algorithm, one can compute q-expansions for a basis of S_{2}(Γ_{0}(N)) consisting of cusp forms with rational Fourier coefficients; cf. for example [Stn]. Write ω_{1}, …, ω_{t} for the corresponding holomorphic 1-forms on X, where for convenience we denote by t = dim_{C} S_{2}(Γ_{0}(N)) the genus of X.

Define η_{i} = uω_{i}, which is a differential of the second kind by the residue theorem, and let B = {ω_{1}, ..., ω_{t}, η_{1}, ..., η_{t}} ⊂ H_{dR}^{1}(X/Q) be the corresponding set of cohomology classes. A simple application of Riemann-Roch shows the following.

Lemma 4.2.1. The set B is basis for H_{dR}^{1}(X/Q) whenever ∞ is not a Weierstrass point on X and u has a pole of order t + 1 (i.e., the smallest possible) at ∞.

Proof. Since ∞ is not a Weierstrass point on X, we may assume that ord_{∞}(ω_{i}) = i−1, and thus ord_{∞}(η_{i}) = i−t−2. For any differential of the second kind ω′, we can find a linear combination of η_{1}, …, η_{t} and dh for an appropriate rational function h having the same principal part as ω′. Thus the difference is holomorphic, and lies in the span of {ω_{1}, …, ω_{t}}. □

Remark 4.2.2. By a result of Ogg [O], the cusp ∞ is not a Weierstrass point when the level N is prime, or more generally when N = pM for prime p and an integer M ≥ 1 such that X_{0}(M) has genus zero and p ∤ M.

When ∞ is a Weierstrass point, there is a rational function with a single pole at ∞ of order ≤ g(X). When u is taken to be such a function, then the set B will never be a basis. Indeed, since ∞ is a Weierstrass point, there exists a holomorphic differential form ω with order of vanishing ≥ g(X) at ∞. Then uω is still holomorphic, and thus lies in the span of {ω_{1}, …, ω_{t}}. But uω also is in the span of {η_{1}, …, η_{t}} by definition of the η_{i}, giving rise to a linear dependence relation. Hence, in order for B to be a basis, it is necessary for u to have a pole at ∞ of order greater than the order of vanishing at ∞ of any holomorphic differential.

This lemma is not strictly necessary for the computation, but rather serves to guarantee its success in certain cases. Even if the hypotheses of the lemma do not hold (for example, if ∞ is not a Weierstrass point but u has a pole of order > t + 1), the set B may still be a basis of H_{dR}^{1}(X/Q), and almost always is at levels < 200. Moreover this can be checked easily in any particular example by computing the matrix for the Poincaré pairing.

Given one basis B for H_{dR}^{1}(X/Q) — for example, one computed as above — it is then a matter of linear algebra to produce a better basis that is adapted to the action of the Hecke algebra. Note that the usual formula for the action of the Hecke algebra T_{0} on holomorphic modular forms in terms of q-expansions extends to weakly holomorphic modular forms (i.e., meromorphic 1-forms on X with possible poles only at the cusps), and preserves the subspace of differentials regular on Y. In particular one can compute the action of T_{0} on any 1-form representing an element of H_{dR}^{1}(X/Q). Using q-series for the elements of the basis B, we can thus write down the matrix [T_{p}] ∈ Mat_{2t×2t}(Q) that describes the action of any T_{p} ∈ T_{0}. After identifying H_{dR}^{1}(X/Q) ≈ Q^{2t} via the basis B, by finding the eigenspaces of finitely many such matrices one can write down Q-bases for each isotypic component of H_{dR}^{1}(X/Q). As is shown in [Stn], the Hecke algebra T_{0} is generated as a Z-module by T_{i} for 1 ≤ i ≤ \frac{m}{6} − \frac{m−1}{N}, where m = [Γ(1) : Γ_{0}(N)]. This gives an upper bound on the number of matrices needed, although in practice considerably fewer are necessary. Using these it is simple linear algebra to produce the desired symplectic basis ω_{g,1}, …, ω_{g,2k} for each isotypic component H_{dR}^{1}(X/Q)[g].

4.3. Modular units and η-products. The preceding discussion raises the question of how to compute the q-expansion about ∞ of a rational function u used to write down an initial choice of basis B for H_{dR}^{1}(X/Q).

Recall that a modular unit (for Γ_{0}(N)) is a modular function u ∈ Q(X)^{×} whose associated divisor is supported on the cusps of X = X_{0}(N). Denote by U the multiplicative group of modular units.

Definition 4.3.1. The eta group U_{η} is the subgroup of Q(X)^{×} of rational functions of the form

u(q) = λ ∏_{0<d|N} η(q^{d})^{r_{d}},

where λ ∈ Q^{×}, η(q) = q^{1/24} ∏_{n>0}(1 − q^{n}) is the classical eta function and {r_{d}}_{d|N} is a collection of integers satisfying the following conditions.

i. ∑_{d|N} r_{d} = 0, ii. ∏_{d|N} d^{r_{d}} ∈ Q^{×} is a square, iii. (n_{d}) := A_{N} · (r_{d}) is a vector (indexed by divisors d of N) of integers divisible by 24, where A_{N} is the σ(N) × σ(N)-matrix whose entry indexed by (d, d′) is \frac{N·(d,d′)^{2}}{dd′(d′,N/d′)}.

Work of Newman and Ligozat shows that such functions are indeed modular units on X; that is, U_{η} ⊂ U. In fact more is true:

Proposition 4.3.2. Q ⊗_{Z} U_{η} = Q ⊗_{Z} U.

Proof. It easy to see that the set \{\frac{a}{d} : d | N, a ∈ (Z/(d, N/d)Z)^{×}\} ⊂ P^{1}(Q) is a complete set of representatives of the cusps of X. The subspace Q ⊗ U_{η} ⊂ Q ⊗ U coincides with Q ⊗ U′, where U′ ⊂ U consists of modular units that have the same valuation at any two cusps a/d, a′/d with the same denominator; cf. [G, Prop. 2]. This implies the proposition in light of the next lemma, since an element u ∈ U ⊂ Q(X) has the same valuation at any two Galois-conjugate cusps. □

Lemma 4.3.3. Let d|N. Then the cusp 1/d is rational if and only if (d, N/d) = 1. More generally, the Galois orbit of the cusp 1/d is Gal (Q̄/Q) · \frac{1}{d} = \{\frac{a}{d} : a ∈ (Z/(d, N/d)Z)^{×}\}.

Proof of the lemma. We prove the first statement using the results of [Stv, §1.3]. Namely, it is known that the cusps of X are rational over Q(ζ_{N}), and the Galois action of Gal (Q(ζ_{N}))/Q) ≃ (Z/NZ)^{×} can be described explicitly as follows [Stv, Thm. 1.3.1]: given b ∈ (Z/NZ)^{×}, let τ_{b} be the automorphism of Q(ζ_{N}) that sends ζ_{N} ↦ ζ_{N}^{b}. If a ∈ Z is chosen so that ab ≡ 1 (mod N) then τ_{b} sends the cusp \frac{1}{d} to \frac{1}{ad} = \frac{a}{d}. Hence the Galois orbit of \frac{1}{d} is

\{\frac{a}{d} : a ∈ (Z/NZ)^{×}\},

and it can be shown by an elementary argument that this set of cusps is equal to the image of

\{\frac{a}{d} | a ∈ (Z/(d, N/d)Z)^{×}\}

in Γ_{0}(N)\P^{1}(Q). □

By the Riemann-Roch theorem, there exist nonconstant rational functions on X that are regular away from ∞. The proposition implies that an integer power of such a function belongs to the subgroup U_{η} ⊂ U, which yields the following.

Corollary 4.3.4. There exists an eta product u ∈ U_{η} that is regular away from ∞. □

It is thus possible to compute the rational function u required in the compation of a basis for H_{dR}^{1}(X) as an eta product.

A practical approach to finding the vector (r_{d})_{d|N} giving rise to the u we seek is to apply a linear programming algorithm: one minimizes the pole order −n_{N} of u at ∞ subject to the criteria of Newman-Ligozat in Definition 4.3.1 and the condition that the orders n_{d} of u at other cusps are non-negative.

Remark 4.3.5. To determine the complexity of the algorithm described in this paper (see §4.8), it is necessary to bound effectively (as a function of N) the order of the pole at ∞ of the eta quotient u in Corollary 4.3.4. This can be done by examining the proof of Corollary 4.3.4. By the Riemann-Roch theorem, there is a nonconstant rational function w on X that is regular on Y and has a pole of order ≤ t = genus(X) at ∞. From the formula for genus(X) as a function on N, one can thus extract the bound −ord_{∞}(w) = O(N log log N); cf. for example [CWZ]. (We adopt the convention that unless decorated with a subscript, an expression O(−) denotes a bound with an absolute implied constant.) An examination of the proof of [G, Prop. 2], which was invoked to show Proposition 4.3.2, actually gives the more precise result that w^{μ} belongs to U_{η} for an integer μ = O(det A_{N}). Combining this with the explicit formula [G, Prop. 1] for det A_{N}, one deduces the estimate −ord_{∞}(u) = O(N^{2σ_{0}(N)+2}), where σ_{0}(N) denotes the number of positive divisors of N.

4.4. Computing the Poincaré dual γ_{f} of ω_{f}. Assume that {γ^{(j)}} is a collection of elements of Γ_{0}(N) with small lower-left entries cN, whose homology classes γ_{0}^{(j)} generate H_{1}(X^{an}, Z). By a brute-force search it is straightforward to find such elements γ^{(j)} in practice. (For small N, often one need take c no greater than 2 or 3.)

For any m ∈ H_{1}(X^{an}, C), write η_{m} for the Poincaré dual of c; conversely, for any differential η of the second kind on X, let m_{η} ∈ H_{1}(X^{an}, C) denote the Poincaré dual of its cohomology class. We normalize the Poincaré duality isomorphism so that it is characterized by the property

(4.4.1)  ⟨η_{m}, η⟩ = ∫_{m} η.

The vector space H_{1}(X^{an}, C) is also equipped an intersection product, which is related to the Poincaré pairing by the formula

(4.4.2)  m · m_{η} = \frac{1}{2πi}⟨η_{m}, η⟩.

The homology of X also admits a natural action of the Hecke algebra, compatible with the action on cohomology via Poincaré duality. For any normalized eigenform f ∈ S_{2}(Γ_{0}(N)) and any m ∈ H_{1}(X^{an}, C), write m^{f} ∈ H_{1}(X^{an}, C)[f] for the projection of m onto the f-isotypic component of homology. Similarly, for η ∈ H_{dR}^{1}(X/Q) write η^{f} for its projection onto the f-isotypic component.

We can assume that via the method described above a symplectic basis

S = {ω_{f,1}, …, ω_{f,ℓ}, η_{f,1}, …, η_{f,ℓ}}

for H_{dR}^{1}(X/Q)[f] has already been computed, where ⟨ω_{f,i}, η_{f,j}⟩ = δ_{i,j}.

Lemma 4.4.1. Fix γ_{1}, γ_{2} ∈ Γ_{0}(N) and let m_{1}, m_{2} ∈ H_{1}(X^{an}, Z) denote the corresponding homology classes on X. For any normalized eigenform f ∈ S_{2}(Γ_{0}(N)), we have

m_{1}^{f} · m_{2}^{f} = \frac{1}{2πi} ∑_{i=1}^{ℓ} I(ω_{f,i}; m_{1})I(η_{f,i}; m_{2}) − I(ω_{f,i}; m_{2})I(η_{f,i}; m_{1}),

where ω_{f} = 2πif(z)dz is the 1-form corresponding to f.

Proof. Let η_{k} = η_{m_{k}} and write η_{k}^{f} = ∑ c_{i}^{(k)}ω_{f,i} + ∑ d_{i}^{(k)}η_{f,i}. Then we compute m_{1}^{f} · m_{2}^{f} = \frac{1}{2πi}⟨η_{1}^{f}, η_{2}^{f}⟩ = ∑_{i} \frac{1}{2πi}(c_{i}^{(1)}d_{i}^{(2)} − c_{i}^{(2)}d_{i}^{(1)}) = \frac{1}{2πi} ∑_{i}(I(ω_{f,i}; γ_{1})I(η_{f,i}, γ_{2}) − I(η_{f,i}; γ_{1})I(ω_{f,i}; γ_{2})).

□

Using (4.4.1), (4.4.2), and Lemma 4.4.1, we can compute the Poincaré dual γ_{f} of ω_{f}. Let m_{1}, …, m_{2ℓ} be modular symbols giving rise to a basis of H_{1}(X^{an}, Z)[f], which can be computed using a modular symbols algorithm (cf. [Stn]). In particular, if f is new, then ℓ = 1. Write A for the matrix (m_{i} ·m_{j})_{1≤i,j≤2ℓ}, which can be computed using Lemma 4.4.1, and let v be the column vector (m_{i} · m_{ω_{f}})_{i=1}^{2ℓ}, which can be computed using (4.4.2) in conjunction with (4.4.1). Then the vector A^{−1}v gives the coefficients expressing m_{ω_{f}} as a linear combination of m_{1}, …, m_{2ℓ}. These coefficients can then easily be converted to an expression for γ_{f} as a linear combination of {γ_{0}^{(j)}}.

4.5. Computing the adjustments ∫_{γ_{f}} α. Write the homology class γ_{f} Poincaré dual to ω_{f} as

γ_{f} = ∑ β_{j}γ_{0}^{(j)}

for β_{j} ∈ C and homology classes γ_{0}^{(j)} whose lifts γ^{(j)} to Γ_{0}(N) have small lower-left entries cN. Let ω and η be differentials of the second kind, at least one of which is regular at ∞. Using the methods described so far, we are already able to compute

z_{ω,η} := ∑_{j} β_{j} ∫_{τ_{0}}^{γ^{(j)}τ_{0}} ωF_{η}.

We stress that the value of z_{ω,η} depends on τ_{0} and the choices we made in representing γ_{f}. It is simply one part of the iterated integral J_{ω,η}(γ_{f}) = ∫_{γ_{f}} ω · η − α_{ω,η}, which is independent of these choices. In this section, we describe a method for computing J_{ω,η}(γ_{f}) − z_{ω,η}. This amounts to computing the q-expansion of α_{ω,η}, for then we have

J_{ω,η}(γ_{f}) − z_{ω,η} = − ∑_{j} β_{j} ∫_{τ_{0}}^{γ^{(j)}τ_{0}} α_{ω,η}.

Recall that the defining property of α_{ω,η} is that its principal part at ∞ agrees with that of ωF_{η} on X̃, modulo dq/q; i.e., their difference has at worst logarithmic poles. However, note that since ∫_{γ_{f}} λ = 0 for exact 1-forms λ, we may replace α_{ω,η} by any cohomologous 1-form. The cohomology class of α_{ω,η} is determined by the data ⟨λ_{i}, α_{ω,η}⟩, where λ_{1}, …, λ_{2t} (for t the genus of X) form a basis of H_{dR}^{1}(X/Q), so it suffices to compute these values of the Poincaré pairing.

We can choose λ_{1}, …, λ_{t} to be holomorphic. In this case, we can compute

⟨λ_{i}, α_{ω,η}⟩ = res_{∞}(F_{λ_{i}} · α_{ω,η}) = res ˜∞(F_{λ_{i}} · F_{η} · ω),

where the second equality holds because res_{∞}(F_{λ_{i}} · α_{ω,η}) depends only on

pp_{∞}(α_{ω,η}) mod \frac{d q}{q} = pp ˜∞(ωF_{η}).

Lemma 4.5.1. Let λ_{1}, …, λ_{t} ∈ H^{0}(X, Ω_{X/Q}^{1}) be a basis of regular 1-forms on X. Then α ∈ H_{dR}^{1}(X/Q) lies in the subspace H^{0}(X, Ω_{X/Q}^{1}) if and only if ⟨λ_{i}, α⟩ = 0 for all 1 ≤ i ≤ t.

Proof. The subspace H^{0}(X, Ω_{X/Q}^{1}) ⊆ H_{dR}^{1}(X) is isotropic for the Poincaré pairing because the pairing can be computed using residues. For dimension reasons, it is maximal isotropic, and the lemma follows. □

By the lemma, if ⟨λ_{i}, α⟩ = ⟨λ_{i}, α′⟩ for i = 1, …, t, then α − α′ is cohomologous to a regular 1-form. Since α_{ω,η} is only well-defined modulo H^{0}(X, Ω_{X/Q}^{1}), it follows that we can choose ⟨λ_{i}, α_{ω,η}⟩ for i = t + 1, …, 2t arbitrarily. For convenience, we choose ⟨λ_{i}, α_{ω,η}⟩ = 0 for i = t + 1, …, 2t. Define the matrix B = (⟨λ_{i}, λ_{j}⟩)_{i,j=1}^{2t} and the vector

w = (⟨λ_{i}, α_{ω,η}⟩)_{i=1}^{2t} = (res ˜∞(F_{λ_{1}}F_{η}ω), ⋯, res ˜∞(F_{λ_{t}}F_{η}ω), \overbrace{0, …, 0}^{t}).

It then follows by elementary linear algebra that the vector B^{−1}w yields the coefficients of an expression for α_{ω,η} as a linear combination of λ_{1}, …, λ_{2t}.

4.6. Computing the coefficients c_{ij}. Using the methods described so far, we may compute all the integrals ∫_{γ_{f}} (ω · η − α_{ω,η}) occuring in (3.2.2). The last input needed to evaluate (3.2.2) is the set of coefficients c_{ij}^{n} appearing in that formula. Using Lemma 3.2.1, these coefficients can be computed using the matrix A_{n} which expresses the right action of T_{n} on H_{dR}^{1}(X/Q)[g] with respect to the basis ω_{g,1}, …, ω_{g,2k}, and the matrix B = (⟨ω_{g,i}, ω_{g,j}⟩)_{1≤i,j≤2k}, which by our choice of basis is the standard symplectic matrix. In section 4.2, we gave a method for computing A_{n} using the action of T_{n} on q-expansions. However, this only works if gcd(n, N) = 1, so we must resort to other methods. We exploit the fact that the action of T_{n} on H_{1}(X^{an}, C)[g] is readily computable using modular symbols; see [Stn] for de! tails. Recall that we have a Hecke-equivariant duality

H_{dR}^{1}(X^{an}, C)[g] × H_{1}(X^{an}, C)[g] → C

given by the integration pairing ⟨ω, λ⟩ = ∫_{γ} λ; here the Hecke-equivariance means that ⟨T_{n}ω, α⟩ = ⟨ω, T_{n}α⟩. Using modular symbols and the techniques of [Stn] one can compute the matrix B_{n} of T_{n} acting on H_{1}(X^{an}, C)[g] on the left, with respect to a basis m_{1}, …, m_{2k}. Write Q = (⟨ω_{i}, m_{j}⟩)_{i,j=1}^{2k}, which can be computed efficiently via the method explained in §4.1

(using an appropriate basis {m_{j}} derived from the generators γ_{0}^{(j)} for H_{1}(X^{an}, Z) discussed above). Then it is straightforward linear algebra to show that A_{n} = QB_{n}Q^{−1}.

4.7. Computing the denominator d_{g,n}. The final ingredient to be computed is the denominator d_{g,n}, or the smallest positive integer such that d_{g,n}T_{g,n} ∈ T_{Z}. This can be accomplished by computing a Z-basis for the (Z-finite free) Hecke algebra T_{Z} as a subring of M_{2t}(Q), where t is the genus of X_{0}(N), by identifying T with an algebra of endomorphisms of the (2t)-dimensional Q-vector space of cuspidal modular symbols of weight 2 and level N. As T_{Z} is generated as an abelian group by T_{i} for 1 ≤ i ≤ \frac{m}{6} − \frac{m−1}{N}, where m = [Γ(1) : Γ_{0}(N)], this is a finite computation. Once T_{Z} has been computed it is a simple matter to find the matrix representation of T_{g,n} and compute the smallest d_{g,n} such that d_{g,n}T_{g,n} ∈ T_{mathbfZ}.

For more details on modular symbols and generating the Hecke algebra, see [Stn].

4.8. Remarks on complexity. The complexity of the computations we have described is primarily determined by the number n_{D} of Fourier coefficients required to compute J_{ω,α}(γ_{f}) to a given number D of digits of accuracy. In this subsection we sketch a method for obtaining a bound on n_{D} in terms of N.

4.8.1. Write the Fourier expansion of u as

u(τ) = ∑_{n≥−n_{0}} c_{n}q^{n}, q = e^{2πiτ} , τ ∈ ℌ.

Let the principal part of u at ∞ be

pp_{∞}(u) = ∑_{−n_{0}≤−m≤0} \frac{d_{m}}{m} q^{−m}, d_{m} = mc_{−m}.

In [BO], Bringmann and Ono prove an exact formula for the Fourier coefficients of harmonic Maass forms, of which weakly holomorphic modular functions such as u are examples. To avoid introducing unnecessary notation, we state only the very special case of their result applicable to our situation. We remark that long ago Rademacher used the circle method to prove a similar exact formula for the coefficients of the j-function [R], and a modification his argument would probably yield a simpler and more direct proof of the special case we require. Using [BO, Thm. 1.1], one can express the coefficients c_{n}, n > 0 in terms of the coefficents d_{m}, the order-1 I-Bessel function I_{1}(z), and the Kloosterman sum

K(−m, n, c) := ∑_{\begin{matrix}0<v<c \\ (v,c)=1 \end{matrix}} exp (\frac{2πi}{c}(nv + mv̄)) ,

where v̄ denotes the multiplicative inverse of v modulo c. Namely, loc. cit. yields the formula

(4.8.1) c_{n} = 2π ∑_{−n_{0}≤−m≤0} d_{m} ∑_{\begin{matrix}c>0 \\ c≡0 (mod N) \end{matrix}}(\frac{m}{n})^{1/2} \frac{K_{0}(−m, n, c)}{c}I_{1} (\frac{4π\sqrt{|mn|}}{c}) ,  n > 0.

By Remark 4.3.5, we have

(4.8.2)  n_{0} = −ord_{∞}(u) ≤ A_{1}N^{2σ_{0}(N)+2},

where the constant A_{1} is absolute and σ_{0} denotes the divisor function. We can trivially bound the numbers d_{m} as follows. Let ξ_{r}(x) = re^{2πix} for 0 < r < 1 and set y = −\frac{1}{2π} log r > 0. The Cauchy integral formula applied to the meromorphic function U(q) = ∑ c_{n}q^{n} of q in the unit disk gives

\frac{d_{m}}{m} = \frac{1}{2πi} ∫_{ξ_{r}} \frac{U(q)}{q^{m+1}} d q = e^{2πmy} ∫_{0}^{1} u(x + iy)e^{−2πinx} d x.

Taking y = 1, say, we thus have

|d_{m}| ≤ me^{2πm} ∫_{0}^{1} |u(x + i)| d x = me^{2πim} ∫_{0}^{1} ∏_{d|N} |η(dx + id)|^{r_{d}} d x,

where η is the Dedekind eta function and (r_{d})_{d|N} is the vector giving rise to u. Recall that η(z) is a nonvanishing holomorphic function on ℌ, and satisfies |η(z +1)| = |η(z)|. Thus there is an absolute upper bound B for max(|η(z)|, |η(z)|^{−1}) on the strip {1 − ϵ ≤ Im(z) ≤ N + ϵ} ⊂ ℌ. It follows that

|d_{m}| ≤ me^{2πm}B^{∑ |r_{d}|}.

To control this quantity we must bound both B and ∑ |r_{d}| in terms of N.

As regards B, note that since η(z) is holomorphic on ℌ^{∗}, the upper bound for |η(z)| on the strip of interest poses no problem; it is a constant independent of N. To find a lower bound for |η(z)| on our strip, we can make use of the Euler formula

η(z) = e^{πiz/12} (1 + ∑_{m≥1} q^{\frac{3m^{2}−m}{2}} + q^{\frac{3m^{2}+m}{2}} ) , q = e^{2πiz}.

From this it is straightforward to deduce a bound |η(z)|^{−1} = O(e^{πN/12}), for z in the strip in question.

To handle ∑ |r_{d}|, we recall some results from numerical analysis. Let A be an invertible σ ×σ matrix of real numbers. We denote by ||A||_{2} the quantity sup_{x∈R^{σ},|x|=1} |Ax| and by ||A||_{max}

the maximum of the absolute values of the matrix entries a_{ij}. A result in numerical linear algebra [Mal, Thm. 2.1] asserts that when A has integer entries, one can control its so-called “condition number” and obtain the estimate

||A^{−1}||_{2} ≤ σ^{2+σ/2}||A||_{max}^{σ−1}.

We apply this result to the matrix A_{N} from Definition 4.3.1 with σ = σ_{0}(N). By Newman and Ligozat’s result, the vector r = (r_{d})_{d|N} and the vector n = (24ord_{a/d}(u))_{d|N} (where a/d is any cusp of X_{0}(N) of denominator d) satisfy r = A_{N}^{−1}n. Moreover since u has a pole of order n_{0} at infinity and no other poles, the fact that ÷(u) has degree 0 implies |n| = O(n_{0}\sqrt{σ_{0}(N)}). Finally, the formula for the entry of A_{N} indexed by (d, d′) entails ||A_{N}||_{max} ≤ N^{3}. We thus have, using (4.8.2),

\sqrt{∑_{d|N} |r_{d}|} ≤ \sqrt{∑_{d|N} |r_{d}|^{2}} = |r| ≤ ||A_{N}^{−1}||_{2}|n| ≤ σ_{0}(N)^{\frac{σ_{0}(N)+5}{2}} n_{0}N^{3(σ_{0}(N)−1)} = O(N^{\frac{11σ_{0}(N)+1}{2}} ).

Consequently, for an absolute constant C we have

|d_{m}| ≪ me^{2πm}e^{πN ∑ |r_{d}|/12} ≤ me^{2πm}e^{CN^{11σ_{0}(N)+2}}
≤ n_{0}e^{2πn_{0}}e^{CN^{11σ_{0}(N)+2}}
= O(n_{0} exp(2πN^{2σ_{0}(N)+2} + CN^{11σ_{0}(N)+2}))

From this we deduce

(4.8.3)  |d_{m}| = O(n_{0} exp(A_{0}N^{11σ_{0}(N)+2})),

for an absolute constant A_{0}.

From (4.8.1) and (4.8.3), standard estimates for Kloosterman sums, and asymptotics for the I-Bessel function, one obtains by the method of [BrPh, §§5.1-2] the estimate

c_{n} = O(N^{5/4}n_{0}^{7/4}n^{−3/4} exp(A_{0}N^{11σ_{0}(N)+2} + N^{−1}4π\sqrt{nn_{0}})).

In light of (4.8.2) this yields

(4.8.4)  c_{n} = O(n^{−3/4} exp(A_{2}N^{11σ_{0}(N)+2} + \frac{4π\sqrt{A_{1}}}{N} N^{σ_{0}(N)+1}\sqrt{n})),

for absolute constants A_{1}, A_{2}.

4.8.2. The coefficients c_{n} determine the Fourier coefficients of the 1-forms occuring in the formula (3.2.2) for P_{g,f,n}. Unfortunately the relationship is indirect, as the construction of the 1-forms ω_{g,i} and α_{ω_{g,i},ω_{g,j}} involves multiplying u against a basis of cusp forms for Γ_{0}(N) and then performing a lot of linear algebra. By Deligne’s proof of the Ramanujan-Petersson conjecture, the cusp forms have nth coefficient of size O_{δ}(n^{\frac{1}{2} +δ}). It follows that nth Fourier coefficient of an element of the basis B for H_{dR}^{1}(X/Q) computed in §4.2 has size at most

O(P (n) exp(A_{2}N^{11σ_{0}(N)+2} + \frac{4π\sqrt{A_{1}}}{N} N^{σ_{0}(N)+1}\sqrt{n})),

for absolute constants A_{1}, A_{2} and a universal polynomial P (n). To compute the 1-forms ω_{g,i} and α_{ω_{g,i},ω_{g,j}}, linear algebra operations are performed on this basis, which spans a vector space of dimension genus(X) = O(N log log N). It thus seems likely that a careful analysis of the linear algebra operations performed would yield a bound

(4.8.5)  O(Q(n) exp(A_{3}N^{11σ_{0}(N)+2} + \frac{4π\sqrt{A_{1}}}{N} N^{σ_{0}(N)+1}\sqrt{n})),

for the nth Fourier coefficient of any 1-form integrated in the course of computing (3.2.2). Here A_{1}, A_{3} are absolute constants and Q(X) is a universal polynomial independent of N.

Suppose λ is such a 1-form (on X or X̃), and consider the problem of integrating the pullback of λ to ℌ^{∗} along a path from τ_{1} to τ_{2}. By the method explained in §4.1, we can assume that imτ_{1}, imτ_{2} ≥ (c_{N}^{∗} · N)^{−1}, where c_{N}^{∗} · N is the largest of the lower-left entries of the elements γ^{(j)} ∈ Γ_{0}(N) introduced at the beginning of §4.4. Recall that these consisted of a collection of elements that span H_{1}(X^{an}, Z) and have lower-left entries as small as possible. We do not know how to bound c_{N}^{∗} in terms of N, although in practice it seems to be very small.

If the Laurent expansion for λ about ∞ (or a lift of ∞ to X̃) is λ = ∑ a_{λ}(n)\frac{d q}{q}, then setting τ_{j} = x_{j} + iy_{j} for j = 1, 2 (where y_{j} ≥ (c_{N}^{∗}N)^{−1}), we have

∫_{τ_{1}}^{τ_{2}} λ = ∑_{n≫−∞} \frac{a_{λ}(n)}{n}(e^{2πinx_{2}}e^{−2πny_{2}} − e^{2πinx_{1}}e^{−2πny_{1}}).

Our problem is to determine n_{D} such that the tails of these sums are bounded by the requisite precision, say 10^{−D}. It clearly suffices to take for n_{D} any m such that

S(m) := ∑_{n≥m} n^{−1}|a_{λ}(n)|e^{−2πny} ≤ 10^{−D}.

Granting (4.8.5), we have

S(m) ≪ ∑_{n≥m} n^{−1}Q(n) exp(A_{3}N^{11σ_{0}(N)+2} + \frac{4π\sqrt{A_{1}}}{N} N^{σ_{0}(N)+1}\sqrt{n} − \frac{2πn}{Nc_{N}^{∗}}).

For a suitable constant A_{4}, we can assume

n^{−1}Q(n) exp(\frac{4π\sqrt{A_{1}}}{N} N^{σ_{0}(N)+1}\sqrt{n}) ≤ exp(\frac{A_{4}}{N} N^{σ_{0}(N)+1}\sqrt{n}).

Thus

S(m) ≪ exp(A_{3}N^{11σ_{0}(N)+2}) ∑_{n≥m} exp(\frac{1}{Nc_{N}^{∗}} (A_{4}c_{N}^{∗}N^{σ_{0}(N)+1}\sqrt{n} − 2πn))

≤ exp(A_{3}N^{11σ_{0}(N)+2}) ∑_{n≥m} e^{−n/Nc_{N}^{∗}} = exp(A_{3}N^{11σ_{0}(N)+2})\frac{e^{−m/Nc_{N}^{∗}}}{1 − e^{−1/Nc_{N}^{∗}}} ,

provided m is large enough that A_{4}c_{N}^{∗}N^{σ_{0}(N)+1}\sqrt{n} − 2πn ≤ −n for n ≥ m. The latter can be ensured by requiring m ≥ A_{5}c_{N}^{∗2}N^{2σ_{0}(N)+2} for an absolute constant A_{5}. It follows that provided (4.8.5) holds, we have the following estimate for n_{D} in terms of D and N:

(4.8.6)  n_{D} = O(max{Nc_{N}^{∗}(D + N^{11σ_{0}(N)+2}), c_{N}^{∗2}N^{2σ_{0}(N)+2}}),

where the implied constant is absolute, as always.

5. Numerical examples

5.1. Example: 37a1. Take N = 37 in the setup of our algorithm. In this setting, the space of regular differentials on X = X_{0}(37) is spanned by ω_{f} and ω_{g}, which are associated to elliptic curves over Q (labeled 37a1 and 37b1 in Cremona’s database) of ranks 1 and 0, respectively. The elliptic curve 37a1 has minimal Weierstrass equation given by

37a1 : y^{2} + y = x^{3} − x,

and its Mordell-Weil group is generated by the point (0, 0).

By computing the periods attached to ω_{f} and ω_{g}, it can be checked that the classes of the matrices

γ_{1} = (\begin{matrix}2 & −1 \\ 37 & −18 \end{matrix}) , γ_{2} = (\begin{matrix}3 & −1 \\ 37 & −12 \end{matrix}) , γ_{3} = (\begin{matrix}5 & 2 \\ 37 & 15 \end{matrix}) , γ_{4} = (\begin{matrix}14 & 3 \\ 37 & 8 \end{matrix})

generate the rational homology of X. These are a “nice” basis for the homology in the sense of the first paragraph of §4.4; that is, the lower left entries are exactly 37 (rather than 37c for |c| > 1), so the integral ∫_{τ}^{γ_{i}τ} λ can be evaluated efficiently for any meromorphic differential 1-form λ on X_{0}(37) or its universal cover regular away form ∞, by the method of §4.1.

To obtain differentials of the second kind representing classes in the deRham cohomology, we consider the elements of the form

η_{1} = u · ω_{f},  η_{2} = u · ω_{g},  u = η(q)^{2}η(q^{37})^{−2} = q^{−3} ∏_{n=1}^{∞}(1 − q^{n})^{2}(1 − q^{37n})^{−2},

where η(q) is the Dedekind eta function. The modular function u is an example of an eta product with its only pole at ∞, as considered in §4.3. It is not hard to check directly (by calculating the Poincaré pairing on all pairs of elements) that the classes of ω_{f}, ω_{g}, η_{1} and η_{2} generate the de Rham cohomology of X; alternatively one could apply Lemma 4.2.1.

After computing the matrix M of the Hecke operator T_{2} acting on H_{dR}^{1}(X_{0}(37)) with respect to the basis ω_{f}, ω_{g}, η_{1}, η_{2}, and then determining the eigenspaces of M, one finds that

η_{f} = \frac{1}{4}(−37ω_{g} + 4η_{1} − 8η_{2}),
η_{g} = \frac{1}{4}(37ω_{f} − 6η_{1} + 10η_{2})

are in the f and g isotypic components of the deRham cohomology respectively. In addition these linear combinations of 1-forms have been chosen so that {ω_{f}, η_{f}} and {ω_{g}, η_{g}} form symplectic bases for the components with respect to the Poincaré pairing.

When one computes the Poincaré dual γ_{f} of ω_{f} as in §4.4, one finds (with our normalization) that it is

γ_{f} = \frac{1}{2πi} (A ([γ_{2}] − [γ_{3}] + [γ_{4}]) − B (−[γ_{1}] + 2[γ_{2}])) .

Here

A ≈ (2.4513893 …)i, B ≈ 2.9934586 …

are certain linear combinations of the periods of f against a basis of H_{1}(X)[f]; see §4.4 for a more exact description.

The method of §4.5 can be used to compute α_{ω_{g},η_{g}}. However, in this case, it is easy to find α_{ω_{g},η_{g}} by inspection. Working with principal parts, one finds that pp_{∞}(ω_{g}F_{η_{g}}) ≡ pp_{∞}(\frac{1}{4}(η_{1} − η_{2})) mod \frac{dq}{q}. Thus we may take α_{ω_{g},η_{g}} = \frac{1}{4}(η_{1} − η_{2}). Integrating this over γ_{f} yields the rational number ∫_{γ_{f}} α_{ω_{g},η_{g}} = −\frac{1}{2} (to many digits of precision).

Since g is a rational newform, then by Remark 3.1.3, we can find all the points P_{g,f,n} by only computing P_{g,f}. According to Remark 3.2.2, this amounts to computing the complex number z_{g,f} := ∫_{γ_{f}} (ω_{g} · η_{g} − η_{g} · ω_{g} − 2α_{ω_{g},η_{g}}). The method in §4.1, coupled with the previous paragraph, yields

z_{g,f} = −0.4093610 … + (1.2256946 …)i.

Let W be the Weierstrass uniformization of E. Then the point W (z_{g,f}) ∈ E(C) does not necessarily lie in E(Q). This is because T_{g} is a rational combination of cycles, and so W (z_{g,f}) is a Q-linear combination of points in E(Q). Thus, the image of W (z_{g,f}) in E(C) ⊗ Q lies in the subspace E(Q)⊗Q. So in order to write P_{g,f} as an element of this space, we must compute the “denominator” of T_{g}. As in §4.7, one can compute using the first few Fourier coefficients of f and g that the idempotent e = (0, 1) ∈ Q × Q ≃ T (⋆) does not belong to T_{Z} ⊂ T but 2e does. Here, the identification (⋆) associates T_{n} ⊗ 1 ∈ T to (a_{n}(f), a_{n}(g)) ∈ Q × Q. By definition, T_{g} corresponds to e as an element of the Hecke algebra, so it’s denominator is 2. Thus, we can write P_{g,f} = W (2z_{g,f}) ⊗ \frac{1}{2} ∈ E(Q) ⊗ Q. One finds that W (2z_{g,f}) agrees with the global point (\frac{1357}{841} : \frac{28888}{24389} : 1) to within 13 digits of accuracy using 350 Fourier coefficients, so we are led to conclude that

P_{g,f} = (\frac{1357}{841} : \frac{28888}{24389} : 1) ⊗ \frac{1}{2} = 12(0 : 0 : 1) ⊗ \frac{1}{2} = 6(0 : 0 : 1) ∈ E(Q) ⊗ Q

5.2. Example: 43a1. Let N = 43 and let E be the elliptic curve labeled 43a1 in Cremona’s database. The modular curve X = X_{0}(43) has genus 3. There are two isotypic components of H_{dR}^{1}(X), one of dimension 2 corresponding to the modular form f that parametrized E, and another of dimension 4 corresponding to a newform g with Fourier coefficients in Q(\sqrt{2}), associated to an abelian surface quotient of J_{0}(43).

In this case, a linear programming algorithm identifies the eta-quotient u that is modular for Γ_{0}(43) of weight 0, holomorphic away from the cusp ∞, and with minimal pole order at ∞, as

u = \frac{η(q)^{4}}{η(q^{43})^{4}} = q^{−7} − 4q^{−6} + 2q^{−5} + 8q^{−4} − 5q^{−3} − 4q^{−2} − 10q^{−1} + 8 + 9q + 14q^{3} + O(q^{4}).

Computing the residue pairing shows that for a basis of cuspforms with rational Fourier coefficients, corresponding to holomorphic 1-forms ω_{f}, ω_{g,1}, ω_{g,2} on X, the collection

ω_{f}, ω_{g,1}, ω_{g,2}, uω_{f}, uω_{g,1}, uω_{g,2}

forms a basis for H_{dR}^{1}(X/Q). By finding the matrices of a few Hecke operators with respect to this basis, one can as in the case N = 37 produce symplectic bases

ω_{f}, η_{f},  and  ω_{g,1}, ω_{g,2}, η_{g,1}, η_{g,2}

for H_{dR}^{1}(X)[f] and H_{dR}^{1}(X)[g] respectively.

We can compute the Poincaré dual γ_{f} and the iterated integrals

∫_{γ_{f}} (ω_{g,i} · ω_{g,j} − α_{ω_{g,i},ω_{g,j}}), ∫_{γ_{f}} (ω_{g,i} · η_{g,j} − α_{ω_{g,i},η_{g,j}}), ∫_{γ_{f}} (η_{g,i} · ω_{g,j} − α_{η_{g,i},ω_{g,j}})

in the same manner as in the case N = 37 with one exception. One simply cannot find α_{ω_{g,i},η_{g,j}} by inspection. No linear combination of our chosen basis has the same principal part as ω_{g,i}F_{η_{g,j}}, however some linear combination is cohomologous to such a form. The techniques from §4.5 can be used to find such a form.

Each T_{g,n} gives rise to an element of End(H_{dR}^{1}(X)[g]) ⊗ Q. The collection of elements arising from T_{g,n}, n ≥ 1 generate a subspace of dimension 2, generated by T_{g,1} and T_{g,2}. Thus, we can effectively compute P_{g,f,n} for all n simply by computing P_{g,f} and P_{g,f,2}. The formula for P_{g,f} is the one given in Remark 3.2.2, so we have

z_{g,f} = ∫_{γ_{f}} (ω_{g,1} · η_{g,1} − η_{g,1} · ω_{g,1} − 2α_{ω_{g,1},η_{g,1}} + ω_{g,2} · η_{g,2} − η_{g,2} · ω_{g,2} − 2α_{ω_{g,2},η_{g,2}})
= −2.0768300 … + (2.7263648 …)i

The Hecke algebra T can be identified with Q×Q(\sqrt{2}) via T_{n}⊗1 ↦ (a_{n}(f), a_{n}(g)). Under this identification, T_{g,1} corresponds to e_{1} = (0, 1), and an examination of the Fourier coefficients of f and g shows that e_{1} does not lie in the image of T_{Z}, but 2e_{1} does. So, we have

P_{g,f} = W (2z_{g,f}) ⊗ \frac{1}{2} = (\frac{11}{49} : −\frac{363}{343} : 1) ⊗ \frac{1}{2} ∈ E(Q) ⊗ Q.

Finding P_{g,f,2} is a little more involved, as we must compute the matrix of T_{2} acting on ω_{g,1}, ω_{g,2}, η_{g,1}, η_{g,2}. Two methods for doing this were discussed in §4.2 and §4.6, and either shows that T_{2}ω_{g,1} = 2ω_{g,2}, T_{2}ω_{g,2} = ω_{g,1}, T_{2}η_{g,1} = −\frac{97997}{132319}ω_{g,2} +η_{g,2} and T_{2}η_{g,2} = \frac{97997}{132319}ω_{g,1} + 2η_{g,1}. So the matrix A_{2} is given by

( \begin{matrix}0 & 2 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & −\frac{97997}{132319} & 0 & 1 \\ \frac{97997}{132319} & 0 & 2 & 0 \end{matrix} ) .

Combining this with Lemma 3.2.1 and remembering that α_{ω_{g,i},ω_{g,j}} = 0 by Remark 2.2.6, we find that

z_{g,f,2} = −\frac{97997}{132319} ∫_{γ_{f}} (ω_{g,1} · ω_{g,2} − ω_{g,2} · ω_{g,1}) + ∫_{γ_{f}} (ω_{g,1} · η_{g,2} − η_{g,2} · ω_{g,1} − 2α_{ω_{g,1},η_{g,2}})
+ 2 ∫_{γ_{f}} (ω_{g,2} · η_{g,1} − η_{g,1} · ω_{g,2} − 2α_{ω_{g,2},η_{g,1}})
= 2.4055874 … − (1.0710898 …)i.

The cycle T_{g,2} corresponds to the element e_{2} = (0, \sqrt{2}) in Q×Q(\sqrt{2}), which belongs to T_{Z} by inspection of the Fourier coefficients of f and g. By evaluating the Weierstrass uniformisation on z_{g,f,s} we find:

P_{g,f,2} = W (z_{g,f,2}) ⊗ 1 = (−1 : 0 : 1) ⊗ 1 ∈ E(Q) ⊗ Q.

5.3. Table. In Table 1 we report some Chow-Heegner points that lie on elliptic curves over Q of rank 1 and conductor < 100. The format of the table is as follows. We list the strong Weil curve E in each isogeny class of rank 1. Let N be the conductor of the curve E in question, and f ∈ S_{2}(Γ_{0}(N)) the newform corresponding to E. The first two columns of the table list the label for E in Cremona’s database and a choice of generator for E(Q), using the canonical minimal Weierstrass equation for E with invariants a_{1}, a_{3} ∈ {0, 1} and a_{2} ∈ {−1, 0, 1}.

The next column indicates a Galois orbit of normalized Hecke eigenforms g ∈ S_{2}(Γ_{0}(N)). The column gives the index of the orbit in question when the orbits are ordered lexicographically by the sequence of traces of T_{1} and T_{p} for all primes p, starting with index 0. In particular, the trace of T_{1} is equal to twice the dimension of S_{2}(N)[g], so the components are sorted first by dimension, and rational newforms are always listed first. As a result, the f-isotypic component, which is omitted from the table, is typically the index 0 component. In parentheses we give the integer N_{g}|N such that g is associated to a newform of level N_{g}.

The fourth column gives an integer n ≥ 1 corresponding to a Hecke operator T_{n}. The table includes [K_{g} : Q]·σ(N/N_{g}) values of n such that T_{g} ·T_{n} form a Q-basis for for the direct factor K_{g} of T. The fifth column lists P_{g,f,n} as an element of E(Q) ⊗_{Z} Q, in terms of the generator P in the second column. The sixth column gives the denominator d_{g,n} defined above.

5.4. Further observations. We conclude with a few remarks.

5.4.1. Table 1 includes several examples where the submodule \underline{P}_{T[g],f} ⊂ \underline{P}_{g,f} ⊂ E(Q) ⊗_{Z} Q vanishes: namely, this occurs when E has label 88a1 and g has index 1, for all g when E is labeled 91b1 or 92b1, and when E has label 99a1 and g has index 2,. This raises the question of whether such vanishing can be explained via Theorem 3.1.4.

At primes p which exactly divide the level N, the local sign depends only on E and can be computed easily from the Fourier expansion of f as ϵ_{p}(g, g, f) = −a_{p}(f) = 1.

Thus, for example, one can check that for the curve labeled 91b1 we have ϵ_{7}(g, g, f) = −a_{7}(f) = −1 and ϵ_{13}(g, g, f) = −a_{13}(f) = −1. Hence Theorem 3.1.4 does not apply; nevertheless, in this case we expect the special value L(f ⊗Sym^{2}(g), 2) not to vanish –a claim which should be in principle numerically verifiable by means of Dokchitser’s L-function calculator (cf. [S^{+}09])– and non-torsion points on E(ℚ) to arise as the Chow-Heegner points associated to the Shimura curve of full level structure on the quaternion algebra of discriminant 91.

Similarly, one can explain the vanishing of \underline{P}_{T[g],f} on the elliptic curve 92b1 because the prime p = 23 exactly divides 92 and ϵ_{23}(g, g, f) = −a_{23}(f) = −1.

As for the examples in Table 1 of torsion Chow-Heegner points at levels 88 and 99, we have ϵ_{11}(g, g, f) = −a_{11}(f) = +1 in both cases, but the local sign at 2 (respectively, at 3) is somewhat more dificult to compute: the reader interested in pursuing this computation will find it helpful to follow the recipe for that sign given by Prasad in [Pr90], which can be approached numerically thanks to the work of Loeffler and Weinstein [LoWe].

In the tables of Stein (see Section 6.3) there is an example of a torsion Chow-Heegner point corresponding to two newforms of level 158. Since this level is squarefree, the hypotheses of Theorem 3.1.4 can be checked, and since the rank of the corresponding elliptic curve (labeled 158b1) is exactly 1, it would appear that L(f ⊗ Sym^{2}(g), 2) = 0 in this example.

5.4.2. Examining Table 1, we see that in each example in which f and g are both newforms of level N (such as when E has label 37a1, or some of the examples when E has label 57a1, 77a1, 99a1, etc.) the index [E(Q) : Zd_{g,1}P_{f,g,1}] is divisible by 4. See Section 6.5.3 for further discussion of this phenomenon.

5.4.3. Let E = E_{f} be the elliptic curve with Cremona label 99a1. As g runs through the Galois orbits of eigenforms of level 99, let the corresponding newforms of levels N_{g}|99 be

g_{99}^{(1)}, g_{99}^{(2)}, g_{99}^{(3)}, g_{33}, g_{11}.

The newforms g_{99}^{(1)} and g_{99}^{(3)} are twists by the quadratic character of conductor 3 of, respectively, g_{33} and g_{11}. Table 1 shows that in E(Q) ⊗ Q we have

P_{f,g_{99}^{(1)},1} = −P_{f,g_{33},1} = −\frac{2}{3}P
P_{f,g_{99}^{(3)},1} = −P_{f,g_{11},1} = \frac{2}{3}P,

where P = (2 : 0 : 1) ∈ E(Q) is a generator. These observations are consistent with the expectation that not only the height (cf. Theorem 3.1.4) but even the exact index [E(Q) : Zd_{g,n}P_{g,f,n}] can be extracted from L-values of the form L(f ⊗ Sym^{2} g, 2) which are invariant under quadratic twists of g.

6. Appendix: Chow-Heegner points associated to pairs of elliptic curves

by William Stein

In this appendix, we consider a special case of Chow-Heegner points that has a simple concrete description due to S. Zhang. Given a pair E, F of nonisogenous elliptic curves, and surjective morphisms φ_{E} : X_{0}(N) → E and φ_{F} : X_{0}(N) → F of curves over ℚ, we associate a rational point P ∈ E(ℚ). We describe a numerical approach to computing P, state some motivating results of Zhang et al. about the height of P, and present a table of data.

6.1. Introduction. Consider a pair E, F of nonisogenous elliptic curves over ℚ and fix surjective morphisms from X_{0}(N) to each curve. We do not assume that N is the conductor of either E or F, though N is necessarily a multiple of the conductor.

\xymatrix{ & X_{0}(N) \ar[dl]_{φ_{E}} \ar[dr]^{φ_{F}} \\ E &  & F}

Let (φ_{E})_{∗} : Div(X_{0}(N)) → Div(E) and φ_{F}^{∗} : Div(F) → Div(X_{0}(N)) be the pushforward and pullback maps on divisors on algebraic curves. Let Q ∈ F (ℂ) be any point, and let

P_{φ_{E},φ_{F} ,Q} = ∑(φ_{E})_{∗}φ_{F}^{∗} (Q) ∈ E(ℂ),

where ∑ means the sum of the points in the divisor using the group law on E, i.e., given a divisor D = ∑ n_{i}P_{i} ∈ Div(E), we have (∑ D) − ∞ ∼ D − deg(D)∞, which uniquely determines ∑ D.

Proposition 6.1.1. The point P_{φ_{E},φ_{F} ,Q} does not depend on the choice of Q.

Proof. The composition (φ_{E})_{∗} ∘ φ_{F}^{∗} induces a homomorphism of elliptic curves

ψ : Pic^{0}(F) = Jac(F) → Jac(E) = Pic^{0}(E).

Our hypothesis that E and F are nonisogenous implies that ψ = 0. We denote by [D] the linear equivalence class of a divisor in the Picard group. If Q′ ∈ F (ℂ) is another point, then under the above composition of maps,

[Q − Q′] ↦ [(φ_{E})_{∗}φ_{F}^{∗} (Q) − (φ_{E})_{∗}φ_{F}^{∗} (Q′)] = [P_{Q} − P_{Q′}].

Thus the divisor P_{Q}−P_{Q′} is linearly equivalent to 0. But F has genus 1, so there is no rational function on F of degree 1, hence P_{Q} = P_{Q′}, as claimed. □

We let P_{φ_{E},φ_{F}} = P_{φ_{E},φ_{F} ,Q} ∈ E(ℂ), for any choice of Q.

Corollary 6.1.2. We have P_{φ_{E},φ_{F}} ∈ E(ℚ).

Proof. Taking Q = O ∈ F (ℚ), we see that the divisor (φ_{E})_{∗} ∘ φ_{F}^{∗} (Q) is rational, so its sum is also rational. □

In the rest of this appendix, we write P_{E,F} = P_{φ_{E},φ_{F}} when E and F are both optimal curves of the same conductor N, and φ_{E} and φ_{F} are as in Section 6.5.

In Section 6.2 we discuss an example in which E and F both have conductor 37. Section 6.3 is about a formula of Yuan-Zhang-Zhang for the height of P_{E,F} in terms of the derivative of an L-function, in some cases. In Section 6.4, we discuss the connection between this appendix and the rest of the paper. The heart of the appendix is Section 6.5, which describes our numerical approach to approximating P_{E,F}. Finally, Section 6.5.2 presents a table of points P_{E,F}.

Acknowledgements: We would like to thank Ralph Greenberg, Ken Ribet, Barry Mazur, Karl Rubin, Shouwu Zhang, and Jon Bober for helpful conversations related to this paper, and especially thank Xinyi Yuan for introducing us to this topic, Henri Darmon for encouraging us to flesh out the details and write it up for publication, and Victor Rotger for clarifying some issues in Section 6.3. We would also like to thank John Cremona for refereeing the corresponding Sage [S^{+}09] code and Cremona, Bas Edixhoven and Samit Dasgupta for contributions to Section 6.5.1.

6.2. Example: N = 37. The smallest conductor curve of rank 1 is the curve E with Cremona label 37a (see [Cre]). The paper [MSD74] discusses the modular curve X_{0}(37) in detail. It gives the affine equation y^{2} = −x^{6} − 9x^{4} − 11x^{2} + 37 for X_{0}(37), and describes how X_{0}(37) is equipped with three independent involutions w, T and S. The quotient of X_{0}(37) by w is E, the quotient by T is an elliptic curve F with F (ℚ) ≈ ℤ/3ℤ and Cremona label 37b, and the quotient by S is the projective line ℙ^{1}.

\xymatrix{ & X_{0}(37) \ar[dl]_{φ_{E}} \ar[d]^{φ_{F}} \ar[dr] \\ E = X/w & F = X/T & ℙ^{1} = X/S}

The maps φ_{E} and φ_{F} have degree 2, by virtue of being induced by an involution. As explained in [MSD74], the fiber over Q = 0 ∈ F (ℚ) contains 2 points:

(1) the cusp [∞] ∈ X_{0}(37)(ℚ), and

(2) the noncuspidal affine rational point (−1, −4) = T (∞) ∈ X_{0}(37)(ℚ). We have φ_{E}([∞]) = 0 ∈ E(ℚ), and [MSD74, Prop. 3, pg. 30] implies that

φ_{F} ((−1, −4)) = (6, 14) = −6(0, −1),

where (0, −1) generates E(ℚ). We conclude that

P_{E,F} = (6, 14)  and  [E(ℚ) : ℤP_{E,F}] = 6.

On [MSD74, pg. 31], they remark: “It would be of the utmost interest to link this index to something else in the theory.”

This remark motivates our desire to compute more examples. Unfortunately, it is very difficult to generalize the above approach directly, since it involves computations with X_{0}(37) and its quotients that rely on explicit defining equations. Just as there are multiple approaches to computing Heegner points, there are several approaches to computing P_{E,F}:

• a Gross-Zagier style formula for the height of P_{E,F} (see Section 6.3),

• explicit evaluation of iterated integrals (see Section 6.4), and

• numerical approximation of the fiber in the upper half plane over a point on F using

a polynomial approximation to φ_{F} (see Section 6.5). This appendix is mainly about the last approach listed above.

6.3. The Formula of Yuan-Zhang-Zhang. Consider a special case of the triple product L-function of [GK92]

(6.3.1)  L(E, F, F, s) = L(E, s) · L(E, Sym^{2}(F), s),

where E and F are elliptic curves of the same conductor N, and all L-functions are normalized so that 1/2 is the center of the critical strip. The following theorem is proved in [YZZ]:

Theorem 6.3.1 (Yuan-Zhang-Zhang). Assume that the local root number of L(E, F, F, s) at every prime of bad reduction is +1 and that the root number at infinity is −1. Then ĥ(P_{E,F}) = (∗) · L′(E, F, F, \frac{1}{2}), where (∗) is nonzero.

Remark 6.3.2. The above formula resembles the Gross-Zagier formula

ĥ(P_{K}) = (∗) · (L(E/ℚ, s) · L(E^{K}/ℚ, s))′|_{s= \frac{1}{2}} ,

where K is a quadratic imaginary field satisfying certain hypotheses.

If one could evaluate L′(E, F, F, \frac{1}{2}), e.g., by applying the algorithm of [Dok04], along with the factor (∗) in the theorem, this would yield an algorithm to compute ±P_{E,F} (mod E(ℚ)_{tor}) when the root number hypothesis is satisfied. Unfortunately, it appears that nobody has numerically evaluated the formula of Theorem 6.3.1 in any interesting cases.

When E and F have the same squarefree conductor N, [GK92, §1] implies that the local root number of L(E, F, F, s) at p is the same as the local root number of E at p; computing the local root number when the level is not square free is more complicated.

Proposition 6.3.3. Assume that E and F have the same squarefree conductor N, that the local root numbers of E at primes p | N are all +1 (equivalently, that we have a_{p}(E) = −1) and that r_{an}(E/ℚ) = 1. Then L(E, Sym^{2} F, \frac{1}{2}) ≠ 0 if and only if ĥ(P_{E,F}) ≠ 0.

Proof. By hypothesis, we have L(E, \frac{1}{2}) = 0 and L′(E, \frac{1}{2}) ≠ 0. Theorem 6.3.1 and the factorization (6.3.1) imply that

ĥ(P_{E,F}) = (∗) · L′(E, \frac{1}{2}) · L(E, Sym^{2} F, \frac{1}{2}),

from which the result follows. □

Section 6.5.2 contains numerous examples in which E has rank 1, F has rank 0, and yet P_{E,F} is a torsion point. The first example is when E is 91b and F is 91a. Then P_{E,F} = (1, 0) is a torsion point (of order 3). In this case, we cannot apply Proposition 6.3.3 since ϵ_{7} = ϵ_{13} = −1 for E. Another example is when E is 99a and F is 99c, where we have P_{E,F} = 0, and ϵ_{3} = ϵ_{11} = +1, but Proposition 6.3.3 does not apply since the level is not square free. Fortunately, we found an example with squarefree level 158 = 2·79: here E is 158b, F is 158d, we have P_{E,F} = 0 and ϵ_{2} = ϵ_{79} = +1, so Proposition 6.3.3 implies that L(E, Sym^{2} F, \frac{1}{2}) = 0.

6.4. Iterated Complex Path Integrals. The body of this paper contains a general approach using iterated path integrals to compute certain Chow-Heegner points, of which P_{E,F} is a specific instance. Comparing our data (Section 6.5.2) with theirs, we find that if E and F are optimal elliptic curves over ℚ of the same conductor N ≤ 100, if e, f ∈ S_{2}(Γ_{0}(N)) are the corresponding newforms, and if P_{f,e,1} ∈ E(ℚ) ⊗_{ℚ} ℚ is the associated Chow-Heegner point (as in the rest of this paper), then 2P_{E,F} = P_{f,e,1}. This is (presumably) a consequence of [DRS].

6.5. A Numerical Approach to Computing P_{E,F}. The numerical approach to computing P that we describe in this section uses relatively little abstract theory. It is inspired by work of Delaunay (see [Del02]) on computing the fiber of the map X_{0}(389) → E over rational points on the rank 2 curve E of conductor 389. We make no guarantee about how many digits of our approximation to P_{E,F} are correct, instead viewing this as an algorithm to produce something that is useful for experimental mathematics only.

Let 𝔥 be the upper half plane, and let Y_{0}(N) = Γ_{0}(N)\𝔥 ⊂ X_{0}(N) be the affine modular curve. Let E and F be nonisogenous optimal elliptic curve quotients of X_{0}(N), with modular parametrization maps φ_{E} and φ_{F}, and assume both Manin constants are 1. Let Λ_{E} and Λ_{F} be the period lattices of E and F, so E ≅ ℂ/Λ_{E} and F ≅ ℂ/Λ_{F}. Viewed as a map [τ] ↦ ℂ/Λ_{E}, we have, using square brackets to denote equivalence classes, that

φ_{E}([τ]) = [∑_{n=1}^{∞} \frac{a_{n}}{n} e^{2πinτ} ] ,

and similarly for φ_{F}, where a_{n} = a_{n}(E) are the L-series coefficients of E (see [Cr, §2.10], which uses the oppositive sign convention). For any positive integer B, define the polynomial

φ_{E,B} = ∑_{n=1}^{B} \frac{a_{n}}{n} T^{n} ∈ ℚ[T],

and similarly for φ_{F,B}.

To approximate P_{E,F}, we proceed as follows. First we make some choices, and after making these choices we run the algorithm, which will either find a “probable” numerical approximation to P_{E,F} or fail.

• y ∈ ℝ_{>0} – minimum imaginary part of points in fiber,

• d ∈ ℤ_{>0} – degree of the first approximation to φ_{F} in Step 1 below,

• r ∈ ℝ_{≠0} – real number specified to b bits of precision that defines Q ∈ ℂ/Λ,

• b′ – bits of precision when dividing points into Γ_{0}(N) orbits, and

• n – number of trials before we give up and output FAIL. We compute P_{E,F,Q} using an approach that will always fail if Q is a ramification point.

Our algorithm will also fail if any points in the fiber over Q are cusps. This is why we do not allow r = 0. One can modify the algorithm to work when Q is an unramified torsion point by using modular symbols and keeping track of images of cusps.

To increase our confidence that we have computed the right point P_{E,F}, we often carry out the complete computation with more than one choice of r.

(1) Low precision roots: Compute all complex double precision roots of the polynomial φ_{F,d} − r. One way to do this is to use “balanced QR reduction of the companion matrix”, as implemented in GSL.^{2} Record the roots that correspond to τ ∈ 𝔥 with ℑ(τ) ≥ y.

(2) High precision roots: Compute B such that if ℑ(τ) ≥ y, then

| ∑_{n=B+1}^{∞} \frac{a_{n}(F)}{n}τ^{n}| < 2^{−b},

where b is the number of bits of precision of r. Summing the tail end of the series and using that |a_{n}| ≤ n (see [GJP^{+}09, Lem. 2.9]), we find that

B = ⌈\frac{log(2^{−(b+1)} · (1 − e^{−2πy_{1}}))}{−2πy}⌉

works. Next, compute the polynomial φ_{F,B} ∈ ℚ[T], and use Newton iteration to refine all roots saved in Step 1 to roots α of f = φ_{F,B} − r ∈ ℝ[T] such that |f(α)| < 2^{−b}. Save those roots that correspond to τ ∈ 𝔥 with ℑ(τ) ≥ y.

(3) Γ_{0}(N)-orbits: Divide the τ’s from Step 2 into Γ_{0}(N)-equivalence classes, testing equivalence to the chosen bit precision b′ ≤ b, as explained in Section 6.5.1. It is easy to efficiently compute the modular degree m_{F} = deg(φ_{F}) (see [Wat02]). If we find m_{F} distinct Γ_{0}(N) classes of points, we suspect that we have found the fiber over [r], so we map each element of the fiber to E using φ_{E} and sum, then apply the elliptic exponential to obtain P_{E,F} to some precision, then output this approximation and terminate. If we find more than m_{F} distinct classes, there was an error in the choices of precision in our computation, so we output FAIL (and suggest either increasing b or decreasing b′).

2GSL is the the GNU scientific library, which is part of Sage [S^{+}09]. Rough timings of GSL for this computation: it takes less than a half second for degree 500, about 5 seconds for degree 1000, about 45 seconds for degree 2000, and several minutes for degree 3000.

(4) Try again: We did not find enough points in the fiber. Systematically replace r by r + mΩ_{F}, for m = 1, −1, 2, −2, …, where Ω_{F} is the least real period of F, then try again going to Step 1 and including the new points found. If upon trying n choices r+mΩ_{F} in a row we find no new points, we output FAIL and terminate the algorithm.

6.5.1. Determining Γ_{0}(N) equivalency. The field of meromorphic functions invariant under Γ_{0}(N) is generated by j(z) and j(Nz), so if two points z_{1} and z_{2} in the upper half plane are equivalent under Γ_{0}(N), then z_{1} and z_{2} are equivalent under SL_{2}(ℤ) and Nz_{1} and Nz_{2} are also equivalent under SL_{2}(ℤ). Because of singularities in the affine curve defined by j(z) and j(Nz), the converse is not true: for example, z_{1} = (−2+i)/5 and z_{2} = (2+i)/5 are equivalent under SL_{2}(ℤ) as are 5z_{1} and 5z_{2}, but z_{1} and z_{2} are not equivalent under Γ_{0}(5). This is why the algorithm we give below must take into account singularities.

Suppose we are given arbitrary z_{1} and z_{2} in the upper half plane. We first find g_{1}, g_{2} ∈ SL_{2}(ℤ) such that w_{i} = g_{i}(z_{i}) is the canonical representative for z_{i} in the standard fundamental domain for SL_{2}(ℤ), as explained in [Cr, §2.14] but using interval arithmetic to avoid rounding errors. If w_{1} ≠ w_{2}, then z_{1} and z_{2} are not equivalent under SL_{2}(ℤ), so they cannot be equivalent under Γ_{0}(N). Thus let w = g_{1}(z_{1}) = g_{2}(z_{2}). The elements of PSL_{2}(ℤ) that send z_{1} to z_{2} are the finitely many elements g_{2}^{−1}Ag_{1}, for A ∈ Stab(w), so we check whether any g_{2}^{−1}Ag_{1} is in Γ_{0}(N). The only elements of the standard fundamental domain for SL_{2}(ℤ) with nontrivial stabilizers are w = i, with stabilizer generated by S ∈ PSL_{2}(ℤ) of order 2, and w = e^{2πi/3} with stabilizer generated by ST, where T corresponds to z ↦ z + 1.

6.5.2. Data. We implemented the above algorithm in Sage [S^{+}09]^{3}. The columns of Tables 2– 5 are as follows. The columns labeled E and F contain Cremona labels for elliptic curves, and those labeled r_{E} and r_{F} contain the corresponding ranks. The column labeled E(ℚ) gives a choice of generators P_{1}, P_{2}, … for the Mordell-Weil group, with r_{E} points of infinite order listed first, then 0, 1 or 2 torsion points listed with a subscript of their order. The column labeled P_{E,F} contains a rational point close to the numerically computed Chow-Heegner point, represented in terms of the generators P_{i} from the column labeled E(ℚ), where P_{1} is the first generator, P_{2} the second, and so on. The columns labeled m_{E} and m_{F} give the modular degrees of E and F. The column labeled ϵ’s contains the local root numbers of L(E, s) at each bad prime. The notes column refers to the notes after the table, which give information about the input parameters needed to compute P_{E,F}:

(1) We used y = 10^{−5} and d = 1500, which takes a few minutes.

(2) We used y = \frac{1}{2} · 10^{−5} and d = 3000, which takes over an hour.

We believe that the values of P_{E,F} are “likely” to be correct, but we emphasize again that they are not proven correct. In the table we give an exact point, but the algorithm computes a numerical approximation P̃_{E,F} to P_{E,F} ∈ E(ℚ). We find what we call P_{E,F} in the table by running through several hundred low height points in E(ℚ) and find the one closest to P̃_{E,F}; in all cases, the coordinates of the point we list are within 10^{−5} of the coordinates of P̃_{E,F}.

The table contains every pair E, F of nonisogenous optimal elliptic curves of the same conductor N ≤ 184 with r_{E} = 1, and most (but not all) with N ≤ 250. It also contains a few additional miscellaneous examples, e.g., with r_{E} = 0 and some of larger conductor with r_{F} = 2. Most rows took only a few seconds to compute, though ones with m_{F} large in some cases took much longer; the total CPU time to compute the entire table was about 8 hours. Unless otherwise noted, we used y = 10^{−4}, d = 500, b′ = 20, and r = 0.1 with 53 bits of precision, as in Section 6.5. We also repeated all computations with r = 0.2, and in every case got the same result.

3See http://trac.sagemath.org/sage_trac/ticket/11975.

6.5.3. Discussion. In Tables 2–5 we always have 2 | [E(ℚ)_{/tor} : ℤP_{E,F}]. In may be possible to prove this in some cases by using that when r_{an}(E) = 1 then the sign in the functional equation for L(E, s) is −1, so at least one nontrivial Atkin-Lehner involution w_{q} acts as +1 on E, which means that the map X_{0}(N) → E factors through X_{0}(N) → X_{0}(N)/w_{q}. Also, there are four cases in which the index [E(ℚ)/_{tor} : ℤP_{E,F}] is divisible by a prime ℓ ≥ 5. They are (106b, 106c, ℓ = 11), (118a, 118d, ℓ = 7), (121b, 121d, ℓ = 7), and (158b, 158c, ℓ = 7). These prime divisors do not appear to have anything to do with the invariants of E and F, individually.

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H. D.: Montreal, Canada

   E-mail address: darmon@math.mcgill.ca

M. D.: Berkeley, California

   E-mail address: mwdaub@math.berkeley.edu

S. L.: Stanford, California

   E-mail address: saml@math.stanford.edu

V. R.: UPC, Barcelona, Spain

   E-mail address: victor.rotger@upc.edu

APPENDIX: CHOW-HEEGNER POINTS ASSOCIATED TO PAIRS OF ELLIPTIC CURVES 35

Table 1. Chow-Heegner points on curves of rank 1 and conductor < 100

{Curve \atop E = E_{f}} generator P ∈ E(Q) g (N_{g}) n P_{g,f,n} d_{g,n}

37a1  (0, −1)  1 (37) 1 −6P  2
43a1  (0, −1)  1 (43) 1  4P  2
2  2P  1
53a1  (0, −1)  1 (53) 1 −2P  2
2 −8P  2
3  4P  2
57a1  (2, 1)  1 (57) 1  \frac{4}{3}P 12
2 (57) 1 −\frac{16}{3}P  3
3 (19) 1 −4P  2
3 −4P  2
58a1  (0, −1)  1 (58) 1  4P  4
2 (29) 1  0  2
2  4P  2
3  4P  2
4  4P  2
61a1  (1, −1)  1 (61) 1 −2P  2
2  4P  2
3 −4P  1
65a1  (−1, 1)  1 (65) 1  P  2
2  P  2
2 (65) 1  P  2
2  3P  2
77a1  (2, 3)  1 (77) 1 \frac{12}{5}P 20
2 (77) 1 −\frac{4}{3}P  6
3 (11) 1 \frac{4}{3}P 6
7 \frac{44}{3}P  6
4 (77) 1 −\frac{12}{5}P 10
2 −4P  2
79a1 (0, 0) 1 (79) 1 −4P 2
2 −4P  2
3 −4P  2
4  0  2
5  0  2
82a1  (0, 0)  1 (82) 1  0  4
3  2P  2
2 (41) 1  2P  2
2  0  2
3  2P  2
4  0  2
5  4P  2
6  2P  2
83a1  (0, 0)  1 (83) 1  0  2
2  2P  2
3  4P  2
4 −4P  2
5 −4P  1
7  0  2

{Curve \atop E = E_{f}} generator P ∈ E(Q) g (N_{g}) n P_{g,f,n} d_{g,n}

88a1 (2, −2) 1 (88) 1 0 16
3  0 16
2 (44) 1  0  8
2  8P  2
3 (11) 1  0  2
2  8P  2
4 16P  1
8 16P  1
89a1 (0, −1) 1 (89) 1 \frac{8}{5}P 5
2 (89) 1  \frac{2}{5}P 10
2 \frac{22}{5}P 10
3 −\frac{16}{5}P 10
4 −\frac{2}{5}P 10
6 \frac{24}{5}P 10
91a1  (0, 0)  1 (91) 1  2P  4
2 (91) 1  2P  4
2 −2P  2
3 (91) 1  4P  4
2  2P  2
3  6P  2
91b1 (−1, 3)  0 (91) 1  0  4
2 (91) 1  0  4
2  0  2
3 (91) 1  0  4
2  0  2
3  0  2
92b1 (1, 1)  1 (92) 1  0  2
2 (46) 1  0 15
2 0 5
3 (23) 1  0 20
2  0  5
3  0  4
4  0  6
6  0  1
8  0  5
99a1  (2, 0)  1 (99) 1 −\frac{2}{3}P 12
2 (99) 1  0 12
3 (99) 1 \frac{2}{3}P 6
4 (33) 1  \frac{2}{3}P 12
3 \frac{8}{3}P 3
5 (11) 1 −\frac{2}{3}P  6
3 −\frac{2}{3}P  3
9 −\frac{22}{3}P  3

36 WILLIAM STEIN

Table 2. Chow-Heegner points associated to pairs of elliptic curves (part 1 of 4)

E  ϵ_{p}’s r_{E}  E(ℚ)  m_{E}  F r_{F} m_{F} P_{E,F} Notes
37a  +  1  (0, −1)  2  37b 0 2  −6P_{1}
37b  −  0  (8, 18)_{3}  2  37a 1 2  P_{1}
57a ++ 1  (2, 1)  4  57c 0 12  8P_{1}
57a ++ 1  (2, 1)  4  57b 0 3  −8P_{1}
57b −+ 0 (7/4, −11/8)_{2}, (1, −1)_{2} 3  57a 1 4  0
57b −+ 0 (7/4, −11/8)_{2}, (1, −1)_{2} 3  57c 0 12  0
57c −+ 0  (2, 4)_{5}  12 57a 1 4  3P_{1}
57c −+ 0  (2, 4)_{5}  12 57b 0 3  P_{1}
58a ++ 1  (0, −1)  4  58b 0 4  8P_{1}
58b −+ 0  (−1, 2)_{5}  4  58a 1 4  3P_{1}
77a ++ 1  (2, 3)  4  77b 0 20 24P_{1}  (1)
77a ++ 1  (2, 3)  4  77c 0 6  −4P_{1}
89a  +  1  (0, −1)  2  89b 0 5  4P_{1}
91a ++ 1  (0, 0)  4  91b 1 4  4P_{1}
91b −− 1  (−1, 3), (1, 0)_{3}  4  91a 1 4  P_{2}
92b −− 1  (1, 1)  6  92a 0 2  0
99a ++ 1  (2, 0), (−1, 0)_{2}  4  99b 0 12 −4P_{1}
99a ++ 1  (2, 0), (−1, 0)_{2}  4  99c 0 12  0
99a ++ 1  (2, 0), (−1, 0)_{2}  4  99d 0 6  2P_{1}
102a + + + 1  (2, −4), (0, 0)_{2}  8 102b 0 16 −8P_{1} (1)
102a + + + 1  (2, −4), (0, 0)_{2}  8 102c 0 24 32P_{1}
106b ++ 1  (2, 1)  8 106a 0 6  −4P_{1}
106b ++ 1  (2, 1)  8 106c 0 48 −88P_{1}
106b ++ 1  (2, 1)  8 106d 0 10 12P_{1}
112a ++ 1  (0, −2), (−2, 0)_{2}  8 112b 0 4  0
112a ++ 1  (0, −2), (−2, 0)_{2}  8 112c 0 8  0
118a ++ 1  (0, −1)  4 118b 0 12 −8P_{1} (1)
118a ++ 1  (0, −1)  4 118c 0 6  4P_{1}
118a ++ 1  (0, −1)  4 118d 0 38 −28P_{1}
121b  +  1  (4, 5)  4 121a 0 6  4P_{1}
121b  +  1  (4, 5)  4 121c 0 6  4P_{1}
121b  +  1  (4, 5)  4 121d 0 24 −28P_{1} (2)
123a −− 1  (−4, 1), (−1, 4)_{5}  20 123b 1 4  0
123b ++ 1  (1, 0)  4 123a 1 20  4P_{1}
124a −− 1  (−2, 1), (0, 1)_{3}  6 124b 0 6  0
128a  +  1  (0, 1), (−1, 0)_{2}  4 128b 0 8  0
128a  +  1  (0, 1), (−1, 0)_{2}  4 128c 0 4  0
128a  +  1  (0, 1), (−1, 0)_{2}  4 128d 0 8  0
129a ++ 1  (1, −5)  8 129b 0 15 −8P_{1}
130a + − − 1  (−6, 10), (−1, 10)_{6}  24 130b 0 8  0
130a + − − 1  (−6, 10), (−1, 10)_{6}  24 130c 0 80  0
135a ++ 1  (4, −8)  12 135b 0 36  0  (1)
136a −− 1  (−2, 2), (0, 0)_{2}  8 136b 0 8  0
138a + + + 1  (1, −2), (−2, 1)_{2}  8 138b 0 16 −16P_{1} (1)
138a + + + 1  (1, −2), (−2, 1)_{2}  8 138c 0 8  −8P_{1}
141a −− 1  (−3, −5)  28 141b 0 12  0
141a −− 1  (−3, −5)  28 141c 0 6  0
141a −− 1  (−3, −5)  28 141d 1 4  0

APPENDIX: CHOW-HEEGNER POINTS ASSOCIATED TO PAIRS OF ELLIPTIC CURVES 37

Table 3. Chow-Heegner points associated to pairs of elliptic curves (part 2 of 4)

E  ϵ_{p}’s r_{E}  E(ℚ)  m_{E}  F r_{F} m_{F} P_{E,F} Notes
141a - -  1  (−3, −5)  28 141e 0 12  0
141d ++ 1  (0, −1)  4 141a 1 28 −12P_{1}
141d ++ 1  (0, −1)  4 141b 0 12  4P_{1}
141d ++ 1  (0, −1)  4 141c 0 6  4P_{1}
141d ++ 1  (0, −1)  4 141e 0 12  4P_{1}
142a - -  1  (1, 1)  36 142b 1 4  0
142a - -  1  (1, 1)  36 142c 0 9  0
142a −− 1  (1, 1)  36 142d 0 4  0
142a −− 1  (1, 1)  36 142e 0 324  0  (2)
142b ++ 1  (−1, 0)  4 142a 1 36  4P_{1}  (1)
142b ++ 1  (−1, 0)  4 142c 0 9  −4P_{1}
142b ++ 1  (−1, 0)  4 142d 0 4  4P_{1}
142b ++ 1  (−1, 0)  4 142e 0 324 8P_{1}  (2)
152a ++ 1  (−1, −2)  8 152b 0 8  0
153a ++ 1  (0, 1)  8 153b 1 16  8P_{1}
153a ++ 1  (0, 1)  8 153c 0 8  8P_{1}
153a ++ 1  (0, 1)  8 153d 0 24  0
153b −− 1  (5, −14)  16 153a 1 8  0
153b −− 1  (5, −14)  16 153d 0 24  0
154a + + + 1  (5, 3), (−6, 3)_{2}  24 154b 0 24 −24P_{1}
154a + + + 1  (5, 3), (−6, 3)_{2}  24 154c 0 16 16P_{1}
155a −− 1 (5/4, 31/8), (0, 2)_{5} 20 155b 0 8  0
155a −− 1 (5/4, 31/8), (0, 2)_{5} 20 155c 1 4  0
155c ++ 1  (1, −1)  4 155a 1 20 −12P_{1}
155c ++ 1  (1, −1)  4 155b 0 8  4P_{1}
156a − + − 1  (1, 1), (2, 0)_{2}  12 156b 0 12  0  (1)
158a −− 1  (−1, −4)  32 158b 1 8  0
158a −− 1  (−1, −4)  32 158c 0 48  0  (1)
158a −− 1  (−1, −4)  32 158d 0 40  0
158a −− 1  (−1, −4)  32 158e 0 6  0
158b ++ 1  (0, −1)  8 158a 1 32 −8P_{1}
158b ++ 1  (0, −1)  8 158c 0 48 −56P_{1} (1)
158b ++ 1  (0, −1)  8 158d 0 40  0
158b ++ 1  (0, −1)  8 158e 0 6  −8P_{1}
160a ++ 1  (2, −2), (1, 0)_{2}  8 160b 0 8  0
162a ++ 1  (−2, 4), (1, 1)_{3}  12 162b 0 6  0
162a ++ 1  (−2, 4), (1, 1)_{3}  12 162c 0 6  0
162a ++ 1  (−2, 4), (1, 1)_{3}  12 162d 0 12  0
170a + − − 1  (0, 2), (1, −1)_{2}  16 170d 0 12  0
170a + − − 1  (0, 2), (1, −1)_{2}  16 170e 0 20  0
171b −− 1  (2, −5)  8 171a 0 12  0
171b −− 1  (2, −5)  8 171c 0 96  0  (1)
171b −− 1  (2, −5)  8 171d 0 32  0
175a −− 1  (2, −3)  8 175b 1 16  0  (1)
175a −− 1  (2, −3)  8 175c 0 40  0  (1)
175b ++ 1  (−3, 12)  16 175a 1 8  16P_{1}
175b ++ 1  (−3, 12)  16 175c 0 40 16P_{1}  (1)
176c −− 1  (1, −2)  8 176b 0 8  0  (1)

38 WILLIAM STEIN

Table 4. Chow-Heegner points associated to pairs of elliptic curves (part 3 of 4)

E  ϵ_{p}’s r_{E}  E(ℚ)  m_{E}  F r_{F} m_{F} P_{E,F} Notes
176c −− 1  (1, −2)  8 176a 0 16  0
176c −− 1  (1, −2)  8 176b 0 8  0  (1)
184a −− 1  (0, 1)  8 184c 0 12  0
184a −− 1  (0, 1)  8 184d 0 24  0
184b ++ 1  (2, −1)  8 184a 1 8  0
184b ++ 1  (2, −1)  8 184c 0 12  0
184b ++ 1  (2, −1)  8 184d 0 24  0
185a ++ 1  (4, −13)  48 185b 1 8  8P_{1}
185a ++ 1  (4, −13)  48 185c 1 6  24P_{1}
185b −− 1  (0, 2)  8 185c 1 6  0
185c ++ 1 (−5/4, 3/8), (−1, 0)_{2} 6 185b 1 8  2P_{1}
189a ++ 1  (−1, −2)  12 189b 1 12 −12P_{1}
189a ++ 1  (−1, −2)  12 189c 0 12 12P_{1}
189b −− 1  (−3, 9), (3, 0)_{3}  12 189a 1 12  0
189b −− 1  (−3, 9), (3, 0)_{3}  12 189c 0 12  0
190a − + − 1  (13, −47)  88 190b 1 8  0
190a − + − 1  (13, −47)  88 190c 0 24  0  (1)
190b + + + 1  (1, 2)  8 190c 0 24 16P_{1}  (1)
192a ++ 1  (3, 2), (−1, 0)_{2}  8 192b 0 8  0
192a ++ 1  (3, 2), (−1, 0)_{2}  8 192c 0 8  0
192a ++ 1  (3, 2), (−1, 0)_{2}  8 192d 0 8  0
196a −− 1  (0, −1)  6 196b 0 42  0  (1)
198a + − − 1 (−1, −4), (−4, 2)_{2} 32 198b 0 32  0  (1)
198a + − − 1 (−1, −4), (−4, 2)_{2} 32 198c 0 32  0
198a + − − 1 (−1, −4), (−4, 2)_{2} 32 198d 0 32  0  (1)
198a + − − 1 (−1, −4), (−4, 2)_{2} 32 198e 0 160  0  (1)
200b −− 1  (−1, 1), (−2, 0)_{2}  8 200c 0 24  0
200b −− 1  (−1, 1), (−2, 0)_{2}  8 200d 0 40  0  (1)
200b −− 1  (−1, 1), (−2, 0)_{2}  8 200e 0 24  0
201a ++ 1  (1, −2)  12 201b 1 12  4P_{1}
201b −− 1  (−1, 2)  12 201a 1 12  0
201c ++ 1  (16, −7)  60 201a 1 12 −24P_{1}
201c ++ 1  (16, −7)  60 201b 1 12  8P_{1}
203b −− 1  (2, −5)  8 203a 0 48  0
203b −− 1  (2, −5)  8 203c 0 12  0
205a −− 1  (−1, 8), (2, 1)_{4}  12 205b 0 16  0
205a −− 1  (−1, 8), (2, 1)_{4}  12 205c 0 8  0
208a −− 1  (4, −8)  16 208c 0 12  0
208a −− 1  (4, −8)  16 208d 0 48  0  (1)
208b ++ 1  (4, 4)  16 208a 1 16  0  (1)
208b ++ 1  (4, 4)  16 208c 0 12  0
208b ++ 1  (4, 4)  16 208d 0 48  0  (1)
212a −− 1  (2, 2)  12 212b 0 21  0
214a −− 1  (0, −4)  28 214b 1 12  0  (1)
214a −− 1  (0, −4)  28 214d 0 12  0
214b ++ 1  (0, 0)  12 214a 1 28 −8P_{1} (1)
214b ++ 1  (0, 0)  12 214d 0 12 −4P_{1}

APPENDIX: CHOW-HEEGNER POINTS ASSOCIATED TO PAIRS OF ELLIPTIC CURVES 39

Table 5. Chow-Heegner points associated to pairs of elliptic curves (part 4 of 4)

E  ϵ_{p}’s r_{E}  E(ℚ)  m_{E}  F r_{F} m_{F} P_{E,F} Notes
214c ++ 1  (11, 10)  60 214a 1 28 −4P_{1} (1)
214c ++ 1  (11, 10)  60 214d 0 12 16P_{1}
214c ++ 1  (11, 10)  60 214b 1 12 12P_{1}  (1)
216a ++ 1  (−2, −6)  24 216b 0 24  0
219a ++ 1  (2, −1)  12 219c 1 60 −12P_{1} (1)
219a ++ 1  (2, −1)  12 219b 1 12 −4P_{1}
216a ++ 1  (−2, −6)  24 216d 0 72  0
219b −− 1 (−3/4, −1/8), (0, 1)_{3} 12 219a 1 12  0
219b −− 1 (−3/4, −1/8), (0, 1)_{3} 12 219c 1 60  0  (1)
219c ++ 1  (−6, 7), (10, −5)_{2}  60 219a 1 12 −12P_{1}
219c ++ 1  (−6, 7), (10, −5)_{2}  60 219b 1 12  4P_{1}
220a − − + 1 (−7, 11), (15, 55)_{6}  36 220b 0 12  0
224a ++ 1  (1, 2), (0, 0)_{2}  8 224b 0 8  0
225a ++ 1  (1, 1)  8 225b 0 40  0  (1)
225e −− 1  (−5, 22)  48 225a 1 8  0  (1)
225e −− 1  (−5, 22)  48 225b 0 40  0  (1)
228b − + − 1  (3, 6)  24 228a 0 18  0
232a ++ 1  (2, −4)  16 232b 0 16  0
234c + + + 1  (1, −2), (−2, 1)_{2}  16 234b 0 48  0  (1)
234c + + + 1  (1, −2), (−2, 1)_{2}  16 234e 0 20  0  (1)
235a −− 1  (−2, 3)  12 235c 0 18  0  (1)
236a −− 1  (1, −1)  6 236b 0 14  0
238a − − + 1 (24, 100), (−8, 4)_{2} 112 238b 1 8  0  (1)
238a − − + 1 (24, 100), (−8, 4)_{2} 112 238c 0 16  0  (1)
238a − − + 1 (24, 100), (−8, 4)_{2} 112 238d 0 16  0  (1)
238b + + + 1  (1, 1), (0, 0)_{2}  8 238a 1 112 12P_{1}  (1)
238b + + + 1  (1, 1), (0, 0)_{2}  8 238c 0 16 −4P_{1} (1)
238b + + + 1  (1, 1), (0, 0)_{2}  8 238d 0 16  4P_{1}  (1)
240c + + + 1  (1, 2), (0, 0)_{2}  16 240a 0 16  0
240c + + + 1  (1, 2), (0, 0)_{2}  16 240d 0 16  0  (1)
243a  +  1  (1, 0)  6 243b 0 9  0  (1)
245a −− 1  (7, 17)  48 245c 1 32  0
246d + + + 1  (3, −6), (4, −2)_{2}  48 246a 0 84 24P_{1}  (1)
446a ++ 1  (4, −6)  24 446d 2 88  0  (2)
446b −− 1  (5, −10)  56 446d 2 88  0  (2)
446d +− 2  -  88 446a 1 12  0  (1)
446d +− 2  -  88 446b 1 56  0  (1)
681a ++ 1  (4, 4)  32 681c 2 96 −24P_{1} (2)