# 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 with .

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

## The Period Map¶

Let be a subgroup of that contains for some , and suppose

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

given by

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

## Abelian Varieties Attached to Newforms¶

Fix a newform , where for some . Let be the -conjugates of , where acts via its action on the Fourier coefficients, which are algebraic integers (since they are the eigenvalues of matrices with integer entries). Let

(1)

be the subspace of cusp forms spanned by the -conjugates of . One can show using the results discussed in Section Atkin-Lehner-Li Theory that the above sum is direct, i.e., that has dimension .

The integration pairing induces a -equivariant homomorphism

from modular symbols to the -linear dual of . Here acts on via , and this homomorphism is -stable by Theorem 1.42. The abelian variety attached to is the quotient

Here , and we include the in the notation to emphasize that these are integral modular symbols. See [Shim59] for a proof that is an abelian variety (in particular, is a lattice, and is equipped with a nondegenerate Riemann form).

When , we can also construct as a quotient of the modular Jacobian , so is an abelian variety canonically defined over .

In general, we have an exact sequence

Remark 10.1

When , the abelian variety has a canonical structure of abelian variety over . Moreover, there is a conjecture of Ribet and Serre in [Rib92] that describes the simple abelian varieties over that should arise via this construction. In particular, the conjecture is that is isogenous to some abelian variety if and only if is a number field of degree . The abelian varieties have this property since embeds in and the endomorphism ring over has degree at most (see [Rib92] for details). Ribet proves that his conjecture is a consequence of Serre’s conjecture [Ser87] on modularity of mod 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 is a simple abelian variety over with a number field of degree and if has good reduction at , then is isogenous to some abelian variety .

Remark 10.2

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

## Extended Modular Symbols¶

In this section, we extend the notion of modular symbols to allows symbols of the form where and are arbitrary elements of .

Definition 10.3

The abelian group sym{esM_k} of extended modular symbols of weight is the -span of symbols , with a homogeneous polynomial of degree with integer coefficients, modulo the relations

and modulo any torsion.

Fix a finite index subgroup . Just as for usual modular symbols, is equipped with an action of , and we define the space of extended modular symbols of weight for to be the quotient

The quotient is torsion-free and fixed by .

The integration pairing extends naturally to a pairing

(2)

where we recall from (?) that denotes the space of antiholomorphic cusp forms. Moreover, if

is the natural map, then respects (2) in the sense that for all and , we have

As we will see soon, it is often useful to replace first by and then by an equivalent sum of symbols such that is easier to compute numerically than .

Let be a Dirichlet character of modulus . If , let .index{} Let sym{esM_k(N,eps)} be the quotient of by the relations , for all , , and modulo any torsion.

## Approximating Period Integrals¶

In this section we assume is a congruence subgroup of that contains for some . Suppose , so and is an integer such that , and consider the extended modular symbol . Let denote the integration pairing from Section Pairing Modular Symbols and Modular Forms. Given an arbitrary cusp form , we have

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

Lemma 10.4

(3)

Proof

In practice we will usually be interested in computing the period map when is a newform. Since is a newform, there is a Dirichlet character such that . The period map then factors through the quotient , so it suffices to compute the period map on modular symbols in .

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

Proposition 10.5

For any , and , we have the following relation in :

Proof

By definition, if is a modular symbol and , then . Thus , so

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

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

(4)

vanishes since is constant and . 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 , and presented as a -expansion to some precision, this algorithm outputs an approximation to the period integral .

1. Write , with , and set in Proposition Proposition 10.5.
2. Replacing by if necessary, we find that the imaginary parts of and are both equal to the positive number .
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 for and were to generate a large subspace of . When and , Manin proved in [Man72] that the map sending to is a surjective group homomorphism. When , the author does not know a similar group-theoretic statement. However, we have the following theorem.

Theorem 10.7

Any element of can be written in the form

for some and Moreover, and can be chosen so that , so the error term (4) vanishes.

Figure 10.1

“Transporting” a transportable modular symbol.

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The modular symbol

can be “transported” to

provided that

The author and Helena Verrill prove this theorem in [SV01]. The condition that the error term vanishes means that one can replace by any 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 must lie in . However, if , then does lie in . It would be interesting to know under what circumstances is generated by symbols of the form with . This sometimes fails for odd; for example, when , the condition implies that has an eigenvector with eigenvalue , and hence is of finite order. When is even, the author can see no obstruction to generating using such symbols.

## Speeding Convergence Using Atkin-Lehner¶

Let . Consider the Atkin-Lehner involution on , which is defined by

Here we take the positive square root if is odd. Then is an involution when is even.

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

and one has that if and , then

If is a Dirichlet character of modulus , then the operator sends to . Thus if , then preserves . In particular, acts on .

The next proposition shows how to compute the pairing under certain restrictive assumptions. It generalizes a result of [Cre97b] to higher weight.

Proposition 10.8

Let be a cusp form which is an eigenform for the Atkin-Lehner operator having eigenvalue (thus and is even). Then for any and any , with the property that , we have the following formula, valid for any :

Here .

Proof

By Proposition Proposition 10.5 our condition on implies that . We describe the steps of the following computation below.

For the first equality, we break the path into three paths, and in the second, we apply the -involution to the first term and use that the action of is compatible with the pairing and that is an eigenvector with eigenvalue . In the following sequence of equalities we combine the first two terms and break up the third; then we replace by and regroup:

A good choice for is , so that . This maximizes the minimum of the imaginary parts of and , which results in series that converge more quickly.

Let . The polynomial

satisfies . We obtained this formula by viewing as the symmetric product of the -dimensional space on which acts naturally. For example, observe that since , the symmetric product of two eigenvectors for is an eigenvector in having eigenvalue . For the same reason, if , there need not be a polynomial such that . One remedy is to choose another so that .

Since the imaginary parts of the terms , and in the proposition are all relatively large, the sums appearing at the beginning of Section Approximating Period Integrals converge quickly if is small. It is important to choose in Proposition Proposition 10.8 with small; otherwise the series will converge very slowly.

Remark 10.9

Is there a generalization of Proposition Proposition 10.8 without the restrictions that and is even?

### Another Atkin-Lehner Trick¶

Suppose is an elliptic curve and let be the corresponding -function. Let be the root number of , i.e., the sign of the functional equation for , so , where . Let be the modular form associated to (which exists by [Wil95, BCDT01]). If , then (see Exercise 10.2). We have

If , then . If , then

(5)

For more about computing with -functions of elliptic curves, including a trick for computing quickly without directly computing , see [Coh93, Section 7.5] and [Cre97a, Section 2.11]. One can also find higher derivatives 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 , where for some . Let be as in (1).

Let be any -linear map with the same kernel as ; we call any such map a rational period mapping associated to . Let be the period mapping associated to the -conjugates of . We have a commutative diagram

Recall from Section Abelian Varieties Attached to Newforms that the cokernel of is the abelian variety .

The Hecke algebra acts on the linear dual

by . Let be the kernel of the ring homomorphism that sends to . Let

Since is a newform, one can show that has dimension . Let be a basis for , so

We can thus compute , hence a choice of . To compute , it remains to compute .

Let denote the space of cusp forms with -expansion in . By Exercise 10.3

is a -vector space of dimension . Let be a basis for this -vector space. We will compute with respect to the basis of dual to this basis. Choose elements with the following properties:

1. Using Proposition Proposition 10.5 or Proposition Proposition 10.8, it is possible to compute the period integrals , , efficiently.
2. The elements and for span a space of dimension (i.e., they span ).

Given this data, we can compute

and

We break the integrals into real and imaginary parts because this increases the precision of our answers. Since the vectors and , , span , we have computed .

Remark 10.10

We want to find symbols satisfying the conditions of Proposition Proposition 10.8. This is usually possible when is very small, but in practice it is difficult when 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 of conductor , 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 . Fortunately, it is now known by [Wil95, BCDT01, TW95] that every elliptic curve over 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 , this algorithm outputs equations for all elliptic curves of conductor , up to isogeny.

1. [Modular Symbols] Compute using Section Explicitly Computing .

2. [Find Rational Eigenspaces] Find the -dimensional eigenspaces in 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 , compute all eigenspaces , where runs through integers with . 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 and exclude them.

3. [Find Newforms] Use Algorithm 9.14 to compute to some precision each newform associated to each eigenspace found in step (2).

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

1. [Period Lattice] Compute the corresponding period lattice by computing the image of , as described in Section Computing the Period Mapping.

2. [Compute ] Let . If , swap and , so . By successively applying generators of , we find an equivalent element in , i.e., and .

3. [-invariants] Compute the invariants and of the lattice using the following rapidly convergent series:

where , where is as in step (b). A theorem of Edixhoven (that the Manin constant is an integer) implies that the invariants and of are integers, so it is only necessary to compute to large precision to completely determine them.

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

5. [Prove Correctness] Using Tate’s algorithm, find the conductor of . If the conductor is not , then recompute and using more terms of and real numbers to larger precision, etc. If the conductor is , compute the coefficients of the modular form attached to the elliptic curve , for . Verify that , where are the coefficients of . If this equality holds, then must be isogenous to the elliptic curve attached to , by the Sturm bound (Theorem 9.18) and Faltings’s isogeny theorem. If the equality fails for some , recompute and to larger precision.

There are numerous tricks to optimize the above algorithm. For example, often one can work separately with and and get enough information to find , up to isogeny (see [Cre97b]).

Once we have one curve from each isogeny class of curves of conductor , we find each curve in each isogeny class (which is another interesting problem discussed in [Cre97a]), hence all curves of conductor . If is an elliptic curve, then any curve isogenous to 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 . See [Cre97a, Section 3.8] for more details.

### Finding Curves: -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 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 , we can find all curves of conductor by letting be the set of prime divisors of .

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

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

1. determine the following finite subgroup of :

2. find all -integral points on certain elliptic curves .

In [CL04], there is one example, where they find all curves of conductor by finding all curves with good reduction outside . They finds curves of conductor that divide into isogeny classes. (Note that .)

### 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 , which implies that the smallest conductor rank curve is composite.

## Exercises¶

Exercise 10.1

Prove Lemma 10.4.

Exercise 10.2

Suppose is a newform and that . Let . Prove that

[Hint: Show that . Then substitute for .]

Exercise 10.3

Let be a power series whose coefficients together generate a number field of degree over . Let be the complex vector space spanned by the -conjugates of .

1. Give an example to show that need not have dimension .
2. Suppose has dimension . Prove that is a -vector space of dimension .

Exercise 10.4

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

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