Computing Periods

This chapter is about computing period maps associated to newforms. We assume you have read Chapters General Modular Symbols and Computing with Newforms and that you are familiar with abelian varieties at the level of [Ros86].

In Section The Period Map we introduce the period map and give some examples of situations in which computing it is relevant. Section Abelian Varieties Attached to Newforms is about how to use the period mapping to attach an abelian variety to any newform. In Section Extended Modular Symbols, we introduce extended modular symbols, which are the key computational tool for quickly computing periods of modular symbols. We turn to numerical computation of period integrals in Section Approximating Period Integrals, and in Section Speeding Convergence Using Atkin-Lehner we explain how to use Atkin-Lehner operators to speed convergence. In Section Computing the Period Mapping we explain how to compute the full period map with a minimum amount of work.

Section All Elliptic Curves of Given Conductor briefly sketches three approaches to computing all elliptic curves of a given conductor.

This chapter was inspired by [Cre97a], which contains similar algorithms in the special case of a newform f=\sum a_n q^n \in S_2(\Gamma_0(N)) with a_n\in\Z.

See also [Dok04] for algorithmic methods to compute special values of very general L-functions, which can be used for approximating L(f,s) for arbitrary s.

The Period Map

Let \Gamma be a subgroup of \SL_2(\Z) that contains \Gamma_1(N) for some N, and suppose

f =\sum_{n\geq 1} a_n q^n\in S_k(\Gamma)

is a newform (see Definition Definition 9.9). In this chapter we describe how to approximately compute the complex period mapping

\Phi_f : \sM_k(\Gamma) \to \C,

given by

\Phi_f(P\{\alpha,\beta\}) = \langle f,\, P\{\alpha,\beta\} \rangle
= \int_{\alpha}^{\beta} f(z) P(z,1) d z,

as in Section Pairing Modular Symbols and Modular Forms. As an application, we can approximate the special values L(f,j), for j=1,2,\ldots, k-1 using (?). We can also compute the period lattice attached to a modular abelian variety, which is an important step, e.g., in enumeration of \Q-curves (see, e.g., [GLQ04]) or computation of a curve whose Jacobian is a modular abelian variety A_f (see, e.g., [Wan95]).

Abelian Varieties Attached to Newforms

Fix a newform f\in S_k(\Gamma), where \Gamma_1(N)\subset \Gamma for some N. Let f_1,\ldots, f_d be the \Gal(\Qbar/\Q)-conjugates of f, where \Gal(\Qbar/\Q) acts via its action on the Fourier coefficients, which are algebraic integers (since they are the eigenvalues of matrices with integer entries). Let

(1)V_f = \C f_1 \oplus \cdots \oplus \C f_d \subset S_k(\Gamma)

be the subspace of cusp forms spanned by the \Gal(\Qbar/\Q)-conjugates of f. One can show using the results discussed in Section Atkin-Lehner-Li Theory that the above sum is direct, i.e., that V_f has dimension d.

The integration pairing induces a \T-equivariant homomorphism

\Phi_f : \sM_k(\Gamma) \ra V_f^* = \Hom_\C(V_f,\C)

from modular symbols to the \C-linear dual V_f^* of V_f. Here \T acts on V_f^* via (\vphi t)(x) = \vphi(tx), and this homomorphism is \T-stable by Theorem 1.42. The abelian variety attached to f is the quotient

A_f(\C) = V_f^*/\Phi_f(\sS_k(\Gamma;\Z)).

Here \sS_k(\Gamma;\Z)=\sS_k(\Gamma), and we include the \Z in the notation to emphasize that these are integral modular symbols. See [Shim59] for a proof that A_f(\C) is an abelian variety (in particular, \Phi_f(\sS_k(\Gamma;\Z)) is a lattice, and V_f^* is equipped with a nondegenerate Riemann form).

When k=2, we can also construct A_f as a quotient of the modular Jacobian \Jac(X_\Gamma), so A_f is an abelian variety canonically defined over \Q.

In general, we have an exact sequence

0 \to \Ker(\Phi_f) \to  \sS_k(\Gamma) \to V_f^* \to A_f(\C) \to 0.

Remark 10.1

When k=2, the abelian variety A_f has a canonical structure of abelian variety over \Q. Moreover, there is a conjecture of Ribet and Serre in [Rib92] that describes the simple abelian varieties A over \Q that should arise via this construction. In particular, the conjecture is that A is isogenous to some abelian variety A_f if and only if \End(A/\Q)\tensor\Q is a number field of degree \dim(A). The abelian varieties A_f have this property since \Q(\ldots, a_n(f),\ldots) embeds in \End(A/\Q)\tensor\Q and the endomorphism ring over \Q has degree at most \dim(A) (see [Rib92] for details). Ribet proves that his conjecture is a consequence of Serre’s conjecture [Ser87] on modularity of mod p odd irreducible Galois representations (see Section Applications of Modular Forms). Much of Serre’s conjecture has been proved by Khare and Wintenberger (not published). In particular, it is a theorem that if A is a simple abelian variety over \Q with \End(A/\Q)\tensor\Q a number field of degree \dim(A) and if A has good reduction at 2, then A is isogenous to some abelian variety A_f.

Remark 10.2

When k>2, there is an object called a Grothendieck motive that is attached to f and has a canonical “structure over \Q “. See [Sch90].

Extended Modular Symbols

In this section, we extend the notion of modular symbols to allows symbols of the form P\{w,z\} where w and z are arbitrary elements of \h^*=\h\union\P^1(\Q).

Definition 10.3

The abelian group sym{esM_k} of extended modular symbols of weight k is the \Z-span of symbols P\{w,z\}, with P\in
V_{k-2} a homogeneous polynomial of degree k-2 with integer coefficients, modulo the relations

P\cdot (\{w,y\} + \{y,z\} +
\{z,w\}) = 0

and modulo any torsion.

Fix a finite index subgroup \Gamma\subset \SL_2(\Z). Just as for usual modular symbols, \esM_k is equipped with an action of \Gamma, and we define the space of extended modular symbols of weight k for \Gamma to be the quotient

\esM_k(\Gamma) = (\esM_k/\langle \gamma x - x : \gamma\in\Gamma, x \in \esM_k\rangle)/\tor.

The quotient \esM_k(\Gamma) is torsion-free and fixed by \Gamma.

The integration pairing extends naturally to a pairing

(2)\left(S_k(\Gamma) \oplus \overline{S}_k(\Gamma)\right)
\to \C,

where we recall from (?) that \overline{S}_k(\Gamma) denotes the space of antiholomorphic cusp forms. Moreover, if

\iota: \sM_k(\Gamma)\rightarrow \esM_k(\Gamma)

is the natural map, then \iota respects (2) in the sense that for all f \in S_k(\Gamma) \oplus \overline{S}_k(\Gamma) and x\in \sM_k(\Gamma), we have

\langle f, x\rangle = \langle f , \iota(x)\rangle.

As we will see soon, it is often useful to replace x\in\sM_k(\Gamma) first by \iota(x) and then by an equivalent sum \sum y_i of symbols y_i\in \esM_k(N,\eps) such that \langle f, \sum y_i \rangle is easier to compute numerically than \langle f, x\rangle.

Let \eps be a Dirichlet character of modulus N. If \gamma=\abcd{a}{b}{c}{d} \in \SL_2(\Z), let \eps(\gamma)=\eps(d).index{\eps(\gamma)} Let sym{esM_k(N,eps)} be the quotient of \esM_k(N,\Z[\eps]) by the relations \gamma(x) -
\eps(\gamma) x, for all x \in \sM_k(N,\Z[\eps]), \gamma \in \Gamma_0(N), and modulo any torsion.

Approximating Period Integrals

In this section we assume \Gamma is a congruence subgroup of \SL_2(\Z) that contains \Gamma_1(N) for some N. Suppose \alp\in \h, so \Im(\alpha)>0 and m is an integer such that 0\leq m\leq k-2, and consider the extended modular symbol X^mY^{k-2-m}\{\alp,\infty\}. Let \langle \cdot, \cdot \rangle denote the integration pairing from Section Pairing Modular Symbols and Modular Forms. Given an arbitrary cusp form f =\sum_{n=1}^{\infty} a_n q^n\in S_k(\Gamma), we have

\Phi_f(X^mY^{k-2-m}\{\alpha,\infty\}) &=
\left\langle f, \, X^mY^{k-2-m}\{\alpha,\infty\}\right\rangle \\
&= \int_{\alpha}^{\infty} f(z)z^m dz \\
&=  \sum_{n=1}^{\oo} a_n \int_{\alpha}^{\infty} e^{2\pi i n z} z^m dz.

The reversal of summation and integration is justified because the imaginary part of \alp is positive so that the sum converges absolutely. The following lemma is useful for computing the above infinite sum.

Lemma 10.4

(3)\int_{\alpha}^{\infty} e^{2\pi i n z} z^m dz
\,\,=\,\, e^{2\pi i n \alpha}
\sum_{s=0}^m \left(
\frac{(-1)^s \alpha^{m-s}}
{(2\pi i n)^{s+1}}
\prod_{j=(m+1)-s}^m j\right).


See Exercise 10.1

In practice we will usually be interested in computing the period map \Phi_f when f \in S_k(\Gamma) is a newform. Since f is a newform, there is a Dirichlet character \eps such that f \in
S_k(N,\eps). The period map \Phi_f : \sM_k(\Gamma) \to \C then factors through the quotient \sM_k(N,\eps), so it suffices to compute the period map on modular symbols in \sM_k(N,\eps).

The following proposition is an analogue of [Cre97a, Prop. 2.1.1(5)].

Proposition 10.5

For any \gamma\in \Gamma_0(N), P\in V_{k-2} and \alp\in\h^*, we have the following relation in \esM_k(N,\eps):

P\{\oo, \gamma(\oo)\}
&=& P\{\alp,\gamma(\alp)\} + (P - \eps(\gamma)\gamma^{-1}P)\{\oo,\alp\}\\
&=& \eps(\gamma)(\gamma^{-1}P)\{\alp, \oo\} - P\{\gamma(\alp),\oo\}.


By definition, if x\in\sM_k(N,\eps) is a modular symbol and \gamma\in\Gamma_0(N), then \gamma{}x=\eps(\gamma)x. Thus \eps(\gamma)\gamma^{-1}x=x, so

P\{\oo, \gamma(\oo)\}
&=& P\{\oo,\alp\} + P\{\alp,\gamma(\alp)\} + P\{\gamma(\alp),\gamma(\oo)\}\\
&=& P\{\oo,\alp\} + P\{\alp,\gamma(\alp)\} + \eps(\gamma)\gamma^{-1}(P\{\gamma(\alp),\gamma(\oo)\})\\
&=& P\{\oo,\alp\} + P\{\alp,\gamma(\alp)\} + \eps(\gamma)(\gamma^{-1}P)\{\alp, \oo\}\\
&=& P\{\alp,\gamma(\alp)\} + P\{\oo,\alp\}  - \eps(\gamma)(\gamma^{-1}P)\{\oo, \alp\}\\
&=& P\{\alp,\gamma(\alp)\} + (P - \eps(\gamma)\gamma^{-1}P)\{\oo,\alp\}.

The second equality in the statement of the proposition now follows easily.

In the case of weight 2 and trivial character, the “error term”

(4)(P - \eps(\gamma)\gamma^{-1}P)\{\oo,\alp\}

vanishes since P is constant and \eps(\gamma)=1. In general this term does not vanish. However, we can suitably modify the formulas found in [Cre97a, 2.10] and still obtain an algorithm for computing period integrals.

Algorithm 10.6

Given \gamma\in\Gamma_0(N), P\in V_{k-2} and f\in S_k(N,\eps) presented as a q-expansion to some precision, this algorithm outputs an approximation to the period integral \langle f, \,P\{\oo, \gamma(\oo)\}\rangle.

  1. Write \gamma=\abcd{\hfill a}{b}{cN}{d}\in\gzero, with a,b,c,d\in\Z, and set \alpha = \frac{-d+i}{cN} in Proposition Proposition 10.5.
  2. Replacing \gamma by -\gamma if necessary, we find that the imaginary parts of \alpha and \gamma(\alpha)=\frac{a+i}{cN} are both equal to the positive number \frac{1}{cN}.
  3. Use (?) and Lemma 10.4 to compute the integrals that appear in Proposition Proposition 10.5.

It would be nice if the modular symbols of the form P\{\oo,\gamma(\oo)\} for P\in V_{k-2} and \gamma\in \Gamma_0(N) were to generate a large subspace of \sM_k(N,\eps)\tensor\Q. When k=2 and \eps=1, Manin proved in [Man72] that the map \Gamma_0(N)\ra H_1(X_0(N),\Z) sending \gamma to \{0,\gamma(0)\} is a surjective group homomorphism. When k>2, the author does not know a similar group-theoretic statement. However, we have the following theorem.

Theorem 10.7

Any element of \sS_k(N,\eps) can be written in the form

\sum_{i=1}^n P_{i}\{\infty,\gamma_i(\infty)\}

for some P_i\in V_{k-2} and \gamma_i\in\gzero. Moreover, P_i and \gamma_i can be chosen so that \sum  P_i = \sum \eps(\gamma_i)\gamma_i^{-1}(P_i), so the error term (4) vanishes.

Figure 10.1

“Transporting” a transportable modular symbol.

\put(75,0){`\gamma \infty`}
\put(140,0){`\beta \infty`}
\put(60,54){`\gamma \alpha`}
\put(100,146){`\beta \alpha`}

The modular symbol

P\{\infty ,\gamma \infty \}+Q\{\gamma \infty ,\beta \infty \}

\hspace{2em}=P\{\infty ,\gamma \infty \}+Q\{\infty ,\beta \infty \}-Q\{\infty ,\gamma \infty \}

can be “transported” to

P\{\alpha,\gamma \alpha\}+Q\{\gamma \alpha,\beta \alpha\},

provided that

P\,+\,Q\,-\,Q=\gamma ^{-1}P\,+\,\beta ^{-1}Q\,-\,\gamma ^{-1}Q.

The author and Helena Verrill prove this theorem in [SV01]. The condition that the error term vanishes means that one can replace \infty by any \alpha in the expression for the modular symbol and obtain an equivalent modular symbol. For this reason, we call such modular symbols transportable, as illustrated in Figure 10.1.

Note that in general not every element of the form P\{\oo,\gamma(\oo)\} must lie in \sS_k(N,\eps). However, if \gamma P = P, then P\{\oo,\gamma(\oo)\} does lie in \sS_k(N,\eps). It would be interesting to know under what circumstances \sS_k(N,\eps) is generated by symbols of the form P\{\oo,\gamma(\oo)\} with \gamma P = P. This sometimes fails for k odd; for example, when k=3, the condition \gamma P = P implies that \gamma\in\gzero has an eigenvector with eigenvalue 1, and hence is of finite order. When k is even, the author can see no obstruction to generating \sS_k(N,\eps) using such symbols.

Speeding Convergence Using Atkin-Lehner

Let w_N = \abcd{0}{-1}{N}{\hfill 0} \in \Mat_2(\Z). Consider the Atkin-Lehner involution W_N on M_k(\Gamma_1(N)), which is defined by

W_N(f) &= N^{(2-k)/2} \cdot f|_{[w_N]_k}\\
&= N^{(2-k)/2} \cdot f\left(-\frac{1}{Nz}\right) \cdot N^{k-1} \cdot (Nz)^{-k}\\
&= N^{-k/2} \cdot z^{-k} \cdot f\left(-\frac{1}{Nz}\right).

Here we take the positive square root if k is odd. Then W_N^2 = (-1)^k is an involution when k is even.

There is an operator on modular symbols, which we also denote W_N, which is given by

W_N(P \{\alpha, \beta\}) &= N^{(2-k)/2} \cdot w_N(P) \{w_N(\alpha), w_N(\beta)\}\\
&=N^{(2-k)/2} \cdot P(-Y,NX) \left\{-\frac{1}{\alpha N}, -\frac{1}{\beta N}\right\},

and one has that if f \in S_k(\Gamma_1(N)) and x\in\sM_k(\Gamma_1(N)), then

\langle W_N(f), x\rangle = \langle f, W_N(x) \rangle.

If \eps is a Dirichlet character of modulus N, then the operator W_N sends S_k(N,\eps) to S_k(\Gamma_1(N),\overline{\eps}). Thus if \eps^2=1, then W_N preserves S_k(N,\eps). In particular, W_N acts on S_k(\Gamma_0(N)).

The next proposition shows how to compute the pairing \langle f, P\{\oo,\gamma(\oo)\} \rangle under certain restrictive assumptions. It generalizes a result of [Cre97b] to higher weight.

Proposition 10.8

Let f \in S_k(N,\eps) be a cusp form which is an eigenform for the Atkin-Lehner operator W_N having eigenvalue w\in \{\pm 1\} (thus \eps^2=1 and k is even). Then for any \gamma\in\Gamma_0(N) and any P\in V_{k-2}, with the property that \gamma P = \eps(\gamma)P, we have the following formula, valid for any \alpha\in\h:

\langle f, P\{\oo,\gamma(\oo)\}\rangle &=
\Bigl\langle f, \quad w \frac{P(Y,-NX)}{N^{k/2-1}}\{w_N(\alp),\oo\}\\
&\quad +
\left(P - w \frac{P(Y,-NX)}{N^{k/2-1}} \right)\left\{i/\sqrt{N},\oo\right\}
-P\left\{\gamma(\alp),\oo\right\} \Bigr\rangle.

Here \ds w_N(\alpha) = -\frac{1}{N\alpha}.


By Proposition Proposition 10.5 our condition on P implies that P\{\oo,\gamma(\oo)\}= P\{\alp,\gamma(\alp)\}. We describe the steps of the following computation below.

&\Bigl\langle f,\quad P\{\alp,\gamma(\alp)\}\Bigr\rangle \\
f,\quad\!\! P\{\alp,i/\sqrt{N}\} + P\{i/\sqrt{N},W(\alp)\}+P\{W(\alp),\gamma(\alp)\}
\right\rangle \\
&=\left\langle f,\quad\!\! w \frac{W(P)}{N^{k/2-1}}
\{W(\alp),i/\sqrt{N}\} + P\{i/\sqrt{N},W(\alp)\}+P\{W(\alp),\gamma(\alp)\}

For the first equality, we break the path into three paths, and in the second, we apply the W-involution to the first term and use that the action of W is compatible with the pairing \langle \,,\, \rangle and that f is an eigenvector with eigenvalue w. In the following sequence of equalities we combine the first two terms and break up the third; then we replace \{ W(\alp), i/\sqrt{N}\} by \{W(\alp),\infty\}+\{\infty,i/\sqrt{N}\} and regroup:

&w \frac{W(P)}{N^{k/2-1}}
\{W(\alp),i/\sqrt{N}\} + P\{i/\sqrt{N},W(\alp)\}+P\{W(\alp),\gamma(\alp)\}\\
&= \left(w \frac{W(P)}{N^{k/2-1}}-P\right)
\{W(\alp),i/\sqrt{N}\} +P\{W(\alp),\oo\} - P\{\gamma(\alp),\oo\}\\
&=  w \frac{W(P)}{N^{k/2-1}}\{W(\alp),\oo\}
+\left(P - w \frac{W(P)}{N^{k/2-1}}\right)\{i/\sqrt{N},\oo\}

A good choice for \alp is \alp=\gamma^{-1}\left(\frac{b}{d}+\frac{i}{d\sqrt{N}}\right), so that W(\alp) = \frac{c}{d}+\frac{i}{d\sqrt{N}}. This maximizes the minimum of the imaginary parts of \alp and W(\alp), which results in series that converge more quickly.

Let \gamma=\abcd{a}{b}{c}{d}\in \Gamma_0(N). The polynomial

P(X,Y) = (cX^2 + (d-a)XY - bY^2)^{\frac{k-2}{2}}

satisfies \gamma(P)=P. We obtained this formula by viewing V_{k-2} as the (k-2)^{th} symmetric product of the 2-dimensional space on which \gzero acts naturally. For example, observe that since \det(\gamma)=1, the symmetric product of two eigenvectors for \gamma is an eigenvector in V_{2} having eigenvalue 1. For the same reason, if \eps(\gamma)\neq 1, there need not be a polynomial P(X,Y) such that \gamma(P)=\eps(\gamma) P. One remedy is to choose another \gamma so that \eps(\gamma)=1.

Since the imaginary parts of the terms i/\sqrt{N}, \alp and W(\alp) in the proposition are all relatively large, the sums appearing at the beginning of Section Approximating Period Integrals converge quickly if d is small. It is important to choose \gamma in Proposition Proposition 10.8 with d small; otherwise the series will converge very slowly.

Remark 10.9

Is there a generalization of Proposition Proposition 10.8 without the restrictions that \eps^2=1 and k is even?

Another Atkin-Lehner Trick

Suppose E is an elliptic curve and let L(E,s) be the corresponding L-function. Let \eps\in\{\pm 1\} be the root number of E, i.e., the sign of the functional equation for L(E,s), so \Lambda(E,s) =\eps\Lambda(E,2-s), where \Lambda(E,s) = N^{s/2} (2\pi)^{-s}\Gamma(s) L(E,s). Let f=f_E be the modular form associated to E (which exists by [Wil95, BCDT01]). If W_N(f) = w f, then \eps = -w (see Exercise 10.2). We have

L(E,1) &= -2\pi \int_{0}^{\infty} f(z)\, dz \\
&= -2\pi i \, \left\langle f,\, \{0,\infty\}\right\rangle\\
&=-2\pi i\, \left\langle f, \{0,i/\sqrt{N}\} + \{i/\sqrt{N}, \infty\}\right\rangle\\
&=-2\pi i\, \left\langle wf, \{w_N(0), w_N(i/\sqrt{N})\}
+ \{i/\sqrt{N}, \infty\}\right\rangle\\
&=-2\pi i\, \left\langle wf, \{\infty , i/\sqrt{N}\}
+ \{i/\sqrt{N}, \infty\}\right\rangle\\
&=-2\pi i\, (w-1) \left\langle f, \{\infty , i/\sqrt{N}\} \right\rangle.\\

If w=1, then L(E,1)=0. If w= -1, then

(5)L(E,1) = 4\pi i \left\langle f, \{\infty , i/\sqrt{N}\} \right\rangle
=  2 \sum_{n=1}^{\infty} \frac{a_n}{n} e^{-2\pi n/\sqrt{N}}.

For more about computing with L-functions of elliptic curves, including a trick for computing \eps quickly without directly computing W_N, see [Coh93, Section 7.5] and [Cre97a, Section 2.11]. One can also find higher derivatives L^{(r)}(E,1) by a formula similar to (5) (see [Cre97a, Section 2.13]). The methods in this chapter for obtaining rapidly converging series are not just of computational interest; see, e.g., [Gre83] for a nontrivial theoretical application to the Birch and Swinnerton-Dyer conjecture.

Computing the Period Mapping

Fix a newform f=\sum a_nq^n\in S_k(\Gamma), where \Gamma_1(N)\subset \Gamma for some N. Let V_f be as in (1).

Let \Theta_f:M_k(\Gamma;\Q) \to V be any \Q-linear map with the same kernel as \Phi_f; we call any such map a rational period mapping associated to f. Let \Phi_f be the period mapping associated to the \Gal(\Qbar/\Q)-conjugates of f. We have a commutative diagram

&          & \Hom_\C(V_f,\C) \\
&     V\ar@{^(->}[ur]^{i_f}

Recall from Section Abelian Varieties Attached to Newforms that the cokernel of \Phi_f is the abelian variety A_f(\C).

The Hecke algebra \T acts on the linear dual

\sM_k(\Gamma;\Q)^* = \Hom (\sM_k(\Gamma), \Q)

by (t \vphi)(x) = \vphi(tx). Let I=I_f\subset \T be the kernel of the ring homomorphism \T\ra \Z[a_2, a_3, \ldots] that sends T_n to a_n. Let

\sM_k(\Gamma;\Q)^*[I] = \{\vphi \in \sM_k(\Gamma;\Q)^* : t \vphi =
0 \text{ all }t \in I\}.

Since f is a newform, one can show that \sM_k(\Gamma;\Q)^*[I] has dimension d. Let \theta_1, \ldots, \theta_d be a basis for \sM_k(\Gamma;\Q)^*[I], so

\Ker(\Phi_f) = \Ker(\theta_1) \oplus \cdots \oplus \Ker(\theta_d).

We can thus compute \Ker(\Phi_f), hence a choice of \Theta_f. To compute \Phi_f, it remains to compute i_f.

Let S_k(\Gamma;\Q) denote the space of cusp forms with q-expansion in \Q[[q]]. By Exercise 10.3

S_k(\Gamma; \Q)[I] = S_k(\Gamma)[I] \cap \Q[[q]]

is a \Q-vector space of dimension d. Let g_1,\ldots,g_d be a basis for this \Q-vector space. We will compute \Phi_f with respect to the basis of \Hom_\Q(S_k(\Gamma;\Q)[I];\C) dual to this basis. Choose elements x_1,\ldots,x_d\in \sM_k(\Gamma) with the following properties:

  1. Using Proposition Proposition 10.5 or Proposition Proposition 10.8, it is possible to compute the period integrals \langle g_i, x_j \rangle, i,j\in\{1,\ldots, d\}, efficiently.
  2. The 2d elements v+\eta(v) and v-\eta(v) for v=\Theta_f(x_1),\ldots,\Theta_f(x_d) span a space of dimension 2d (i.e., they span \sM_k(\Gamma)/\Ker(\Phi_f)).

Given this data, we can compute

i_f (v+\eta(v)) =
2\Re(\langle g_1, x_i\rangle, \ldots, \langle g_d, x_i\rangle)


i_f(v-\eta(v)) =
2i\Im(\langle g_1, x_i\rangle, \ldots, \langle g_d, x_i\rangle).

We break the integrals into real and imaginary parts because this increases the precision of our answers. Since the vectors v_n+\eta(v_n) and v_n-\eta(v_n), n=1,\ldots,d, span \sM_k(N,\eps;\Q)/\Ker(\Phi_f), we have computed i_f.

Remark 10.10

We want to find symbols x_i satisfying the conditions of Proposition Proposition 10.8. This is usually possible when d is very small, but in practice it is difficult when d is large.

Remark 10.11

The above strategy was motivated by [Cre97a, Section 2.10].

All Elliptic Curves of Given Conductor

Using modular symbols and the period map, we can compute all elliptic curves over \Q of conductor N, up to isogeny. The algorithm in this section gives all modular elliptic curves (up to isogeny), i.e., elliptic curves attached to modular forms, of conductor N. Fortunately, it is now known by [Wil95, BCDT01, TW95] that every elliptic curve over \Q is modular, so the procedure of this section gives all elliptic curves (up to isogeny) of given conductor. See [Cre06] for a nice historical discussion of this problem.

Algorithm 10.12

Given N> 0, this algorithm outputs equations for all elliptic curves of conductor N, up to isogeny.

  1. [Modular Symbols] Compute \sM_2(\Gamma_0(N)) using Section Explicitly Computing .

  2. [Find Rational Eigenspaces] Find the 2-dimensional eigenspaces V in \sM_2(\Gamma_0(N))_{\new} that correspond to elliptic curves. Do not use the algorithm for decomposition from Section Decomposing Spaces under the Action of Matrix, which is too complicated and gives more information than we need. Instead, for the first few primes p\nmid N, compute all eigenspaces \Ker(T_p - a), where a runs through integers with -2\sqrt{p} < a  <  2\sqrt{p}. Intersect these eigenspaces to find the eigenspaces that correspond to elliptic curves. To find just the new ones, either compute the degeneracy maps to lower level or find all the rational eigenspaces of all levels that strictly divide N and exclude them.

  3. [Find Newforms] Use Algorithm 9.14 to compute to some precision each newform f=\sum_{n=1}^{\infty} a_n q^n \in \Z[[q]] associated to each eigenspace V found in step (2).

  4. [Find Each Curve] For each newform f found in step (3), do the following:

    1. [Period Lattice] Compute the corresponding period lattice \Lambda=\Z\omega_1 + \Z\omega_2 by computing the image of \Phi_f, as described in Section Computing the Period Mapping.

    2. [Compute \tau] Let \tau=\omega_1/\omega_2. If \Im(\tau)<0, swap \omega_1 and \omega_2, so \Im(\tau)>0. By successively applying generators of \SL_2(\Z), we find an \SL_2(\Z) equivalent element \tau' in \cF, i.e., |\Re(\tau')|\leq 1/2 and |\tau|\geq 1.

    3. [c-invariants] Compute the invariants c_4 and c_6 of the lattice \Lambda using the following rapidly convergent series:

      c_4 &= \left(\frac{2\pi}{\omega_2}\right)^4 \cdot
c_6 &= \left(\frac{2\pi}{\omega_2}\right)^6 \cdot

      where q=e^{2\pi i \tau'}, where \tau' is as in step (b). A theorem of Edixhoven (that the Manin constant is an integer) implies that the invariants c_4 and c_6 of \Lambda are integers, so it is only necessary to compute \Lambda to large precision to completely determine them.

    4. [Elliptic Curve] An elliptic curve with invariants c_4 and c_6 is

      E: \quad y^2 = x^3 -\frac{c_4}{48}x  -\frac{c_6}{864}.

    5. [Prove Correctness] Using Tate’s algorithm, find the conductor of E. If the conductor is not N, then recompute c_4 and c_6 using more terms of f and real numbers to larger precision, etc. If the conductor is N, compute the coefficients b_p of the modular form g=g_E attached to the elliptic curve E, for p\leq \#\P^1(\Z/N\Z)/6. Verify that a_p = b_p, where a_p are the coefficients of f. If this equality holds, then E must be isogenous to the elliptic curve attached to f, by the Sturm bound (Theorem 9.18) and Faltings’s isogeny theorem. If the equality fails for some p, recompute c_4 and c_6 to larger precision.

There are numerous tricks to optimize the above algorithm. For example, often one can work separately with \sM_k(\Gamma_0(N))_{\new}^+ and \sM_k(\Gamma_0(N))_{\new}^- and get enough information to find E, up to isogeny (see [Cre97b]).

Once we have one curve from each isogeny class of curves of conductor N, we find each curve in each isogeny class (which is another interesting problem discussed in [Cre97a]), hence all curves of conductor N. If E/\Q is an elliptic curve, then any curve isogenous to E is isogenous via a chain of isogenies of prime degree. There is an a priori bound on the degrees of these isogenies due to Mazur. Also, there are various methods for finding all isogenies of a given degree with domain E. See [Cre97a, Section 3.8] for more details.

Finding Curves: S-Integral Points

In this section we briefly survey an alternative approach to finding curves of a given conductor by finding integral points on other elliptic curves.

Cremona and others have developed a complementary approach to the problem of computing all elliptic curves of given conductor (see [CL04]). Instead of computing all curves of given conductor, we instead consider the seemingly more difficult problem of finding all curves with good reduction outside a finite set S of primes. Since one can compute the conductor of a curve using Tate’s algorithm [Tat75, Cre97a, Section 3.2], if we know all curves with good reduction outside S, we can find all curves of conductor N by letting S be the set of prime divisors of N.

There is a strategy for finding all curves with good reduction outside S. It is not an algorithm, in the sense that it is always guaranteed to terminate (the modular symbols method above is an algorithm), but in practice it often works. Also, this strategy makes sense over any number field, whereas the modular symbols method does not (there are generalizations of modular symbols to other number fields).

Fix a finite set S of primes of a number field K. It is a theorem of Shafarevich that there are only finitely many elliptic curves with good reduction outside S (see [Sil82, Section IX.6]). His proof uses that the group of S-units in K is finite and Siegel’s theorem that there are only finitely many S-integral points on an elliptic curve. One can make all this explicit, and sometimes in practice one can compute all these S-integral points.

The problem of finding all elliptic curves with good reduction outside of S can be broken into several subproblems, the main ones being

  1. determine the following finite subgroup of K^*/(K^*)^m:

    K(S,m) = \{x \in K^*/(K^*)^m \, : \, m \mid \ord_\p(x) \text{ all } \p\not\in S\};

  2. find all S-integral points on certain elliptic curves y^2=x^3 + k.

In [CL04], there is one example, where they find all curves of conductor N=2^8\cdot 17^2 = 73984 by finding all curves with good reduction outside \{2, 17\}. They finds 32 curves of conductor 73984 that divide into 16 isogeny classes. (Note that \dim S_2(\Gamma_0(N)) = 9577.)

Finding Curves: Enumeration

One can also find curves by simply enumerating Weierstrass equations. For example, the paper [SW02] discusses a database that the author and Watkins created that contains hundreds of millions of elliptic curves. It was constructed by enumerating Weierstrass equations of a certain form. This database does not contain every curve of each conductor included in the database. It is, however, fairly complete in some cases. For example, using the Mestre method of graphs [Mes86], we verified in [JBS03] that the database contains all elliptic curve of prime conductor < 234446, which implies that the smallest conductor rank 4 curve is composite.


Exercise 10.1

Prove Lemma 10.4.

Exercise 10.2

Suppose f \in S_2(\Gamma_0(N)) is a newform and that W_N(f) = w f. Let \Lambda(E,s) = N^{s/2} (2\pi)^{-s} \Gamma(s) L(E,s). Prove that

\Lambda(E,s) = -w \Lambda(E,2-s).

[Hint: Show that \Lambda(f,s) = \int_{0,\infty} f(iy/\sqrt{N})y^{s-1}\, dy. Then substitute 1/y for y.]

Exercise 10.3

Let f=\sum a_n q^n \in \C[[q]] be a power series whose coefficients a_n together generate a number field K of degree d over \Q. Let V_f be the complex vector space spanned by the \Gal(\Qbar/\Q)-conjugates of f.

  1. Give an example to show that V_f need not have dimension d.
  2. Suppose V_f has dimension d. Prove that V_f \cap
\Q[[q]] is a \Q-vector space of dimension d.

Exercise 10.4

Find an elliptic curve of conductor 11 using Section All Elliptic Curves of Given Conductor.