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 .

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**]).

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**].

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.

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

See *Exercise 10.1*

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 .

- Write , with ,
and set in Proposition
*Proposition 10.5*. - Replacing by if necessary, we find that the imaginary parts of and are both equal to the positive number .
- 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.

```
\begin{figure}
\begin{center}
\begin{picture}(230,150)(0,0)
\put(-10,10){\line(1,0){200}}
\put(10,10){\circle*{3}}
\put(10,0){`\infty`}
\qbezier(10,10)(50,40)(75,10)
\put(49,25){\vector(1,0){1}}
\put(45,28){`P`}
\put(75,10){\circle*{3}}
\put(75,0){`\gamma \infty`}
\qbezier(75,10)(130,50)(140,10)
\put(120,30){\vector(1,0){1}}
\put(115,34){`Q`}
\put(140,10){\circle*{3}}
\put(140,0){`\beta \infty`}
\put(20,90){\circle*{3}}
\put(17,96){`\alpha`}
\qbezier(20,90)(50,120)(55,60)
\put(40,100){\vector(1,0){1}}
\put(40,104){`P`}
\put(55,60){\circle*{3}}
\put(60,54){`\gamma \alpha`}
\qbezier(55,60)(70,180)(100,140)
\put(87,150){\vector(1,0){1}}
\put(85,154){`Q`}
\put(100,140){\circle*{3}}
\put(100,146){`\beta \alpha`}
\comment{
\qbezier(10,10)(-10,30)(10,50)
\qbezier(10,50)(40,70)(20,90)
\put(5,45){\vector(2,3){1}}
\qbezier(75,10)(90,30)(75,40)
\qbezier(75,40)(60,50)(55,60)
\put(69,45){\vector(1,-1){1}}
\put(80,35){\vector(-1,2){1}}
\qbezier(140,10)(160,30)(140,50)
\qbezier(140,50)(90,80)(110,90)
\qbezier(110,90)(130,100)(100,120)
\qbezier(100,120)(90,130)(100,140)
\put(110,90){\vector(-1,-1){1}}
}
\put(100,50){\vector(-1,4){10}}
\put(50,40){\vector(-1,4){10}}
\put(140,155){\parbox[t][4in][l]{2in}
{
```

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.

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?

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.

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:

- Using Proposition
*Proposition 10.5*or Proposition*Proposition 10.8*, it is possible to compute the period integrals , , efficiently. - 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].

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.

[Modular Symbols] Compute using Section

*Explicitly Computing*.[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.[Find Newforms] Use

*Algorithm 9.14*to compute to some precision each newform associated to each eigenspace found in step*(2)*.[Find Each Curve] For each newform found in step

*(3)*, do the following:[Period Lattice] Compute the corresponding period lattice by computing the image of , as described in Section

*Computing the Period Mapping*.[Compute ] Let . If , swap and , so . By successively applying generators of , we find an equivalent element in , i.e., and .

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

[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.

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

determine the following finite subgroup of :

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 .)

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.

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 .

- Give an example to show that need not have dimension .
- 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*.