Cuspidal modular symbols are transportable

William A. Stein

Harvard University, USA was@math.harvard.edu http://modular.fas.harvard.edu

Helena A. Verrill

Universität Hannover verrill@math.uni-hannover.de

Abstract

Modular symbols of weight 2 for a congruence subgroup Γ satisfy the identity {α, γ(α)} = {β, γ(β)} for all α, β in the extended upper half plane and γ ∈ Γ. The analogue of this identity is false for modular symbols of weight greater than 2. In this paper we define transportable modular symbols, which are symbols for which an analogue of the above identity holds, and prove that every cuspidal symbol can be written as a transportable symbol. As a corollary, we obtain an algorithm for computing periods of cuspforms.

Introduction

It is well known that modular symbols of weight 2 for a congruence subgroup Γ satisfy the identity {α, γ(α)} = {β, γ(β)} for all α, β in the extended upper half plane and γ ∈ Γ. The analogue of this identity is, in general, false for modular symbols of weight greater than 2. To investigate further, we define transportable modular symbols, which are symbols that can be expressed in such a way that the above identity holds. We then prove that every cuspidal symbol is transportable. As a corollary we obtain an algorithm for computing periods of cuspforms.

In Section 1 we review the definition of modular symbols. In Section 2 we define transportable modular symbols and prove our main theorem. Section 3 contains an application of our transportability result to the computation of periods of modular forms. Finally, Section 4 contains two examples in which we verify the assertion of Theorem 2.4 and apply the period computation algorithm.

Acknowledgement. The authors would like to thank L. Merel and J. Cremona for helpful advice, I. Kiming for improving the proof of Theorem 2.4, and the referee for several helpful remarks.

1 Modular symbols

In Section 1.1 we recall the definition of modular symbols given in [5], then in Section 1.2 we introduce a slight generalization of the definition. Let N and k be positive integers with k ≥ 2, and let ε : Z/NZ → ℂ be a Dirichlet character modulo N.

1.1 Definition

Let M be the abelian group generated by all symbols {α, β} with α, β ∈ P^{1}(Q), modulo the relations {α, β} + {β, γ} + {γ, α} = 0, and modulo any torsion. Let V_{k−2} denote the group of homogeneous polynomials in Z[X, Y] of degree k − 2. Then each element γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, Z) acts on the left on V_{k−2} by

γ(P (X, Y)) = P (dX − bY, −cX + aY),

and on M_{k} = V_{k−2} ⊗ M by

γ(P ⊗ {α, β}) = γ(P) ⊗ {γ(α), γ(β)}.

Fix a Dirichlet character ε : Z/NZ → ℂ, and denote by Z[ε] the ring generated by the image of ε. We also view ε as a homomorphism Γ_{0}(N) → ℂ^{∗} by setting

ε ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ε(d).

The space M_{k}(N, ε) of modular symbols of level N and character ε is the quotient of the Z[ε]-module M_{k} ⊗ Z[ε] by the Z[ε]-submodule generated by γ(x) − ε(γ)x for all x ∈ M_{k}, all γ ∈ Γ_{0}(N), and by any torsion. Denote by P {α, β} the image of P ⊗ {α, β} in M_{k}(N, ε). The Q[ε]-vector space

M_{k}(N, ε; Q) = M_{k}(N, ε) ⊗_{Z} Q

contains M_{k}(N, ε).

Let B be the free abelian group generated by all symbols {α}, for α ∈ P^{1}(Q). Define a left action of SL(2, Z) on B_{k} = V_{k−2} ⊗ B by

γ(P ⊗ {α}) = γ(P) ⊗ {γα}.

The space B_{k}(N, ε) of boundary symbols is the quotient of B_{k} ⊗ Z[ε] by the Z[ε]-submodule generated by γ(x) − ε(γ)x for all x ∈ B_{k}, all γ ∈ Γ_{0}(N), and by any torsion. The subspace S_{k}(N, ε) of cuspidal symbols is the kernel of the map δ : M_{k}(N, ε) → B_{k}(N, ε) given by δ(P {α, β}) = P {β} − P {α}.

When ε = 1 is the trivial character, we will also write M_{k}(Γ_{0}(N)) for M_{k}(N, 1), and similarly for S_{k} and B_{k}.

1.2 Extended modular symbols

It is useful to extend the notion of modular symbols to allows symbols of the form P {z, w} where z and w are arbitrary elements of 𝔥^{∗} = 𝔥 ∪ P^{1}(Q).

Definition 1 (Extended modular symbols). The group \overline{M}_{k} of extended modular symbols is the free abelian group with basis the set of all symbols P {z, w} with z, w ∈ 𝔥^{∗}, subject to the relations P {u, v}+P {v, w}+P {w, u} = 0.

Note that \overline{M}_{k} is of uncountable rank over Z. It is equipped with an action of Γ_{0}(N); we let \overline{M}_{k}(N, ε) be the largest torsion-free quotient of \overline{M}_{k} by the relations γx = ε(γ)x for γ ∈ Γ_{0}(N).

2 Transportable modular symbols

In Section 2.1 we define transportable modular symbols, and we prove an elementary proposition that motivates the definition. Section 2.2, which is the heart of this paper, contains a proof that every cuspidal modular symbol is transportable.

2.1 Definition

Definition 2 (Transportable). A modular symbol is transportable if it can be written in the form

∑_{i=1}^{m} P_{i}{∞, γ_{i}(∞)},

for γ_{i} ∈ Γ_{0}(N) and P_{i} ∈ V_{k−2} with

∑_{i=1}^{m} P_{i}{∞, γ_{i}(∞)} = ∑_{i=1}^{m} P_{i}{α, γ_{i}(α)}

for all α ∈ 𝔥^{∗}, where the equality takes place in \overline{M}_{k}.

When k = 2, the identity {∞, γ(∞)} = {α, γ(α)} holds for any α ∈ 𝔥^{∗}, so in weight 2 there is a plentiful supply of transportable modular symbols. This identity sometimes fails when k > 2, as indicated below.

Proposition 2.1. For any γ ∈ Γ_{0}(N), P ∈ V_{k−2} and α ∈ 𝔥^{∗},

P {∞, γ(∞)} = P {α, γ(α)} + (P − ε(γ)γ^{−1}P){∞, α}
= ε(γ)(γ^{−1}P){α, ∞} − P {γ(α), ∞}.  (1)

In particular,

P {∞, γ(∞)} = P {α, γ(α)} ⇔ P = ε(γ)γ^{−1}P.  (2)

Proof. If x ∈ M_{k}(N, ε) is a modular symbol and γ ∈ Γ_{0}(N) then γx = ε(γ)x, where, as usual, ε is viewed as a homomorphism Γ_{0}(N) → ℂ^{∗} via ε(( \begin{matrix}a & b \\ c & d \end{matrix} )) = ε(d). In particular, ε(γ)γ^{−1}x = x, so

P {∞, γ(∞)} = P {∞, α} + P {α, γ(α)} + P {γ(α), γ(∞)}
= P {∞, α} + P {α, γ(α)} + ε(γ)γ^{−1}(P {γ(α), γ(∞)})
= P {∞, α} + P {α, γ(α)} + ε(γ)(γ^{−1}P){α, ∞}
= P {α, γ(α)} + P {∞, α} − ε(γ)(γ^{−1}P){∞, α}
= P {α, γ(α)} + (P − ε(γ)γ^{−1}P){∞, α}.

The remaining statements of the proposition now follow easily.

Example 2.2. In some cases it is easy to give a formula for symbols that are obviously transportable. Suppose k ≥ 2 is an even integer. If P is a polynomial such that γ(P) = P for some γ ∈ Γ_{0}(N), then P {∞, γ^{−1}(∞)} is transportable. Given γ ∈ Γ_{0}(N), an example of such a P is

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

We found this polynomial by viewing V_{k−2} as the (k − 2)th symmetric product of the 2-dimensional space on which Γ_{0}(N) acts naturally. If γ, which has determinant 1, has eigenvalues α and α^{−1}, then the eigenvalues of the k − 2 fold symmetric product of γ are given by α^{k−2−2j} for 0 ≤ j ≤ k − 2. Although we have not been able to find a counterexample, the authors see no reason to believe that transportable symbols of the form given in this example always span S_{k}(N; Q).

∞

P
γ∞

Q

β∞

α P

γα

Q βα

[figure] [figure]

The modular symbol

P {∞, γ∞} + Q{γ∞, β∞}

= P {∞, γ∞}+Q{∞, β∞}−Q{∞, γ∞}

can be “transported” to

P {α, γα} + Q{γα, βα},

provided that

P + Q − Q = γ^{−1}P + β^{−1}Q − γ^{−1}Q.

Figure 1: “Transporting” a transportable modular symbol.

However, by definition, any transportable symbol can be written in the form ∑_{i=1}^{n} P_{i}{∞, g_{i} ∞} for some finite sequence {g_{1}, …, g_{n}} ⊂ Γ_{0}(N). To find such transportable symbols we just have to find the kernel of the map ⨁_{i=1}^{n}(1 − g_{i}) from ⨁_{i=1}^{n} V_{k−2} to V_{k−2}. This is a simple matter of linear algebra.

2.2 Characterization of transportable modular symbols

Lemma 2.3. A modular symbol in M_{k}(N, ε; Q) is transportable if and only if it can be written in the form ∑_{i=1}^{m} P_{i}{∞, γ_{i}(∞)} with

∑ P_{i} = ∑ ε(γ_{i})γ_{i}^{−1}P_{i}.

Proof. This follows from Proposition 2.1.

Figure 1 illustrates Lemma 2.3 with a trivial-character example.

Theorem 2.4. A modular symbol is transportable if and only if it is cuspidal.

Proof. By Lemma 2.3 every transportable modular symbols is cuspidal, so we must prove that every cuspidal symbol is transportable.

Let I = I_{N,ε} be the ideal in the group ring of Γ_{0}(N) generated by all elements of the form ε(γ) − γ for γ ∈ Γ_{0}(N). Suppose v ∈ S_{k}(N, ε). Use the relation {α, β} = {∞, β} − {∞, α} ∈ M to see that any v is the image of an element ṽ ∈ M_{k} of the form

ṽ = ∑_{β∈Q} P_{β} ⊗ {∞, β} ∈ M_{k}

with only finitely many P_{β} nonzero. For later convenience we set P_{∞} = 0, and take sums over all β ∈ P^{1}(Q). The boundary map δ lifts in a natural way to M_{k} = V_{k−2} ⊗ M, as illustrated.

\xymatrix{ & I(V_{k−2} ⊗ M) \ar[r] \ar[d] & I(V_{k−2} ⊗ B) \ar[d] \\  & V_{k−2} ⊗ M \ar[r]^{δ̃} \ar[d] & V_{k−2} ⊗ B \ar[d] \\ S_{k}(N, ε) \ar@{^{(}->}[r] & M_{k}(N, ε) \ar[r]^{δ} & B_{k}(N, ε)}

Bearing in mind torsion, our assumption that δ(v) = 0 implies that for some nonzero M ∈ Z we have Mδ̃(ṽ) ∈ I(V_{k−2} ⊗ B). So there are Q_{γ,β} ∈ V_{k−2}, for γ ∈ Γ_{0}(N) and β ∈ P^{1}(Q), only finitely β many nonzero, such that

Mδ̃(ṽ) = ∑_{γ,β}(ε(γ) − γ)(Q_{γ,β} ⊗ {β}).

We now use a summation trick.

Mδ̃(ṽ) = M ∑_{β}(P_{β} ⊗ {β} − P_{β} ⊗ {∞})
= ∑_{γ,β}(ε(γ)Q_{γ,β} ⊗ {β} − (γQ_{γ,β}) ⊗ {γβ})
= ∑_{γ,β} ε(γ)Q_{γ,β} ⊗ {β} − (γQ_{γ,γ^{−1}β}) ⊗ {β}
= ∑_{γ,β}(ε(γ)Q_{γ,β} − γQ_{γ,γ^{−1}β}) ⊗ {β}.

This shows that

M ∑_{β}(P_{β} ⊗ {β} − P_{β} ⊗ {∞}) = ∑_{γ,β}(ε(γ)Q_{γ,β} − γQ_{γ,γ^{−1}β}) ⊗ {β}. (3)

Equating terms we deduce that for β ≠ ∞,

MP_{β} = ∑_{γ}(ε(γ)Q_{γ,β} − γQ_{γ,γ^{−1}β}).  (4)

Using this expression for P_{β} and that ε(γ)γ^{−1} acts trivially on M_{k}(N, ε), we find that

Mv = M ∑_{β} P_{β}{∞, β} = ∑_{γ,β}(ε(γ)Q_{γ,β} − γQ_{γ,γ^{−1}β}){∞, β}
= ∑_{γ,β} ε(γ)Q_{γ,β} − ε(γ)γ^{−1}((γQ_{γ,γ^{−1}β}){∞, β})
= ∑_{γ,β} ε(γ)Q_{γ,β}{∞, β} − ε(γ)Q_{γ,γ^{−1}β}{γ^{−1} ∞, γ^{−1}β}
= ∑_{γ,β} ε(γ)Q_{γ,β}{∞, β} − ε(γ)Q_{γ,β}{γ^{−1} ∞, β}
= ∑_{γ,β} ε(γ)Q_{γ,β}{∞, γ^{−1} ∞}.  (5)

Equating coefficients of {∞} in Equation 3, we have

−M ∑_{β} P_{β} = ∑_{γ}(ε(γ)Q_{γ,∞} − γQ_{γ,γ^{−1} ∞}),

which, combining with Equation 4, and recalling that P_{∞} = 0, means that

− ∑_{γ,β≠∞}(ε(γ)Q_{γ,β} − γQ_{γ,γ^{−1}β}) = ∑_{γ}(ε(γ)Q_{γ,∞} − γQ_{γ,γ^{−1} ∞}),

and hence

∑_{γ,β}(ε(γ)Q_{γ,β} − γQ_{γ,β}) = 0.

Using the expression v = −\frac{1}{M} ∑_{β,γ} ε(γ)Q_{γ,β}\{∞, γ^{−1} ∞\} obtained from Equation 5, we see this is the condition for v to be transportable.

Corollary 2.5. Fix α ∈ 𝔥^{∗}. Every element of S_{k}(N, ε) is a sum of modular symbols of the form P {α, γ(α)}.

Proof. Let x ∈ S_{k}(N, ε). Proposition 2.1 implies that x is transportable, so there exists P_{i} and γ_{i} such that

x = ∑ P_{i}{∞, γ_{i}(∞)} = ∑ P_{i}{β, γ_{i}(β)}

for any β ∈ 𝔥^{∗}. Taking β = α proves the corollary.

Remark 2.6.

1. When k = 2, the corollary follows from [4, §1], which asserts that map Γ_{0}(N) → S_{2}(Γ_{0}(N)) = H_{1}(X_{0}(N), Z) sending γ to {α, γ(α)} is a surjective group homomorphism.

2. In Proposition 2.7 below, we will prove more generally that every element of M_{k}(N, ε) is a sum of modular symbols of the form P {α, γ(α)}, as long as we allow α to vary over P^{1}(Q).

2.3 What space do the symbols P {∞, γ(∞)} span?

Suppose N and k are positive integers, with k even.

Definition 3. For any α ∈ P^{1}(Q), let W_{α} denote the subspace of M_{k}(Γ_{0}(N); Q) spanned by symbols of the form P {α, γ(α)}, for P ∈ V_{k−2} and γ ∈ Γ_{0}(N).

Corollary 2.5 draws our attention to W_{∞}. Since W_{∞} contains S_{k}(Γ_{0}(N)), it is natural to ask how much bigger it is. As mentioned in Remark 2.6, when k = 2, Manin proved that for any α ∈ P^{1}(Q), we have W_{α} = W_{∞} = S_{2}(Γ_{0}(N); Q). We now compute W_{α} for any weight k > 2:

Proposition 2.7. Suppose k > 2. Then the space W_{α} is equal to the inverse image under the boundary map δ of the one-dimensional subspace V_{k−2}{α} ⊂ B_{k}(Γ_{0}(N); Q). Hence dim W_{α} = dim S_{k}(Γ_{0}(N); Q) + 1 and M_{k}(Γ_{0}(N); Q) = ∑_{α∈P^{1}(Q)} W_{α}.

Proof. In [5, §1.4], Merel shows that V_{k−2}{α} has dimension 1 (see the proof of [5, Prop. 4, §1.4]), and that P (X, Y)\{\frac{u}{v}\} is nonzero if P (u, v) ≠ 0.

Corollary 2.5 implies that W_{α} contains the kernel S_{k}(Γ_{0}(N)) of the boundary map δ. It thus suffices to show that δ(W_{α}) = V_{k−2}{α}. For P ∈ V_{k−2} and γ ∈ Γ_{0}(N), we have

δ(P {α, γ(α)}) = P {γ(α)} − P {α} = (γ^{−1}P − P){α} ∈ V_{k−2}{α},

so δ(W_{α}) ⊂ V_{k−2}\{α\}. For γ = ( \begin{matrix}1 & 0 \\ N & 1 \end{matrix} ) ∈ Γ_{0}(N), we have

δ(X^{k−3}Y {α, γ(α)}) = (γ^{−1}(X^{k−3}Y) − X^{k−3}Y){α}
= (X^{k−3}(NX + Y) − X^{k−3}Y){α}
= NX^{k−2}{α}.

If α ≠ 0, then, as mentioned above, X^{k−2}{α} ≠ 0. (If α = 0, use XY^{k−3} and γ = ( \begin{matrix}1 & N \\ 0 & 1 \end{matrix} ) instead.) Because there is a nonzero element in δ(W_{α}) and V_{k−2}\{α\} has dimension 1, it follows that δ(W_{α}) = V_{k−2}{α}. The final claim of the proposition is true because B_{k}(Γ_{0}(N); Q) = ∑_{α∈P^{1}(Q)} V_{k−2}{α}.

Corollary 2.8. Fix α ∈ P^{1}(Q). Then W_{α} = M_{k}(Γ_{0}(N); Q) if and only if N = 1.

Proof. When N = 1, γ can be any element of SL_{2}(Z), so the assertion is clear. Next suppose that W_{α} = M_{k}(Γ_{0}(N); Q). If k = 2 then by [4, §1] W_{α} = S_{k}(Γ_{0}(N); Q), so N = 1 since there is always a weight 2 Eisenstein series when N > 1. Next suppose that k > 2. By [5, Prop. 5, §1.4], δ is surjective and by [5, Prop. 5, §1.4] the dimension of the image of δ equals #Γ_{0}(N)\P^{1}(Q). Combining Proposition 2.7 with our assumption that W_{α} = M_{k}(Γ_{0}(N); Q) implies that #Γ_{0}(N)\P^{1}(Q) = 1, so N = 1, as claimed.

3 Application to computing periods of newforms

The authors were led to introduce transportable modular symbols in order to study the error term (P − ε(γ)γ^{−1}P){∞, α} of Equation 1 of Proposition 2.1 in the context of computing periods of newforms. There are many ways to compute periods of newforms, but we hope that the method given below will be of value in some contexts.

Section 3.1 contains an algorithm for computing periods that relies on Theorem 2.4. We present a potentially more efficient method in Section 3.2.

3.1 An algorithm for computing periods

Let f = ∑ a_{n}q^{n} ∈ S_{k}(N, ε) be a cuspform and let x ∈ M_{k}(N, ε) be a modular symbol. Then ⟨f, x⟩ (see [5, §1.5]) is a linear combination of integrals of the form

⟨f, X^{m}Y^{k−2−m}{α, ∞}⟩ = 2πi ∫_{α}^{i∞} f(z)z^{m}dz,  (6)

where α ∈ 𝔥^{∗} and the integer m satisfies 0 ≤ m ≤ k − 2. If α ∈ 𝔥 then the imaginary part of α is positive, so

2πi ∫_{α}^{i∞} f(z)z^{m}dz = ∑_{n≥1} a_{n}c_{n},

where

c_{n} = 2πi ∫_{α}^{i∞} z^{m}e^{2πinz}dz.

The reversal of summation and integration is justified because the sum converges absolutely. We compute the c_{n} using the following formula, which we obtain using repeated integration by parts.

Lemma 3.1.

∫_{α}^{i∞} e^{2πinz}z^{m}dz = e^{2πinα} ∑_{s=0}^{m} \{\frac{(−1)^{s}α^{m−s}}{(2πin)^{s+1}} · ∏_{j=(m+1)−s}^{m} j\} .

If α has large imaginary part, the c_{n} will rapidly converge to 0 as n → ∞. However, the reversal of summation and integration above need not be valid when α is a real number, so for computational purposes we are led to express periods in terms of integrals with end points that are in 𝔥. When k = 2 this is easy because of the identity {∞, γ(∞)} = {α, γ(α)}, which is valid for any α ∈ 𝔥^{∗}. However, this identity can fail when k > 2; the failure is made precise in Proposition 2.1.

Since we can take the real part of α to be greater than 0, each of the terms on the right hand side of Equation 1 can be computed using the sum given by Lemma 3.1.

We showed in Section 2 that every cuspidal modular symbol can be expressed as a sum of symbols of the form P {∞, γ(∞)}. Periods of modular symbols of this form can then be computed using the following algorithm.

Algorithm 3.2. Given a triple γ ∈ Γ_{0}(N), P ∈ V_{k−2} and g ∈ S_{k}(N, ε) this algorithm computes the period integral ⟨g, P \{∞, γ(∞)\}⟩. Express γ as ( \begin{matrix}a & b \\ cN & d \end{matrix} ) ∈ Γ_{0}(N) and set α = \frac{−d+i}{cN} in Proposition 2.1. Replacing γ by −γ if necessary, we find that the imaginary parts of α and γ(α) = \frac{a+i}{cN} are both equal to the positive number 1/cN. Equation 6 and Lemma 3.1 can now be used to compute the period integrals provided by Proposition 2.1.

3.2 The W_{N}-trick

In this section, in order to obtain a potentially more efficient way of computing periods than Algorithm 3.2, we generalize the method of Cremona [2] to even integer weight k ≥ 2. In Algorithm 3.2, with γ = ( \begin{matrix}a & b \\ cN & d \end{matrix} ), the endpoints of the corresponding integrals have imaginary part \frac{1}{cN}. However, using the following trick one can increase the imaginary part of all endpoints involved to at least \frac{1}{d\sqrt{N}}, which is sometimes a significant improvement.

Recall that the Atkin-Lehner involution W = W_{N} is induced by the matrix ( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} ); it acts on modular forms by sending a cuspform f ∈ S_{k}(N, ε) to the form

f|_{W} (z) = N^{−k/2}z^{−k}f(−1/(Nz)) ∈ S_{k}(N, ε^{−1}).

If f is an eigenvector for W, then necessarily ε = ε^{−1}. For the rest of this section we assume that ε^{2} = 1. Then W acts on M_{k}(N, ε) by

W (P (X, Y)\{α, β\}) = \frac{P (Y, −NX)}{N^{k/2−1}} \{−\frac{1}{Nα}, −\frac{1}{Nβ} \} ,

and this action is compatible with the integration pairing.

Proposition 3.3. Let g ∈ S_{k}(N, ε) be a cuspform that is an eigenform for the Atkin-Lehner involution W having eigenvalue w. Then for any transportable modular symbol ∑_{j=1}^{m} P_{j}{∞, γ_{j}(∞)} with γ_{j} ∈ Γ_{0}(N) and P_{j} ∈ V_{k−2}, we have for any α ∈ 𝔥 the following formula:

⟨g, ∑_{j=1}^{m} P_{j}\{∞, γ_{j}(∞)\}⟩ = ⟨g, ∑_{j=1}^{m} w \frac{P_{j}(Y, −NX)}{N^{k/2−1}} \{W (α), ∞\}

+ ∑_{j=1}^{m} (P_{j} − w \frac{P_{j}(Y, −NX)}{N^{k/2−1}}) \{\frac{i}{\sqrt{N}} , ∞\}

− ∑_{j=1}^{m} P_{j} {γ_{j}(α), ∞}⟩ .

Here W (α) = −1/(Nα).

If γ_{j} = ( \begin{matrix}a_{j} & b_{j} \\ c & d \end{matrix} ), where c and d are fixed integers that do not depend on j, then

⟨g, ∑_{j=1}^{m} P_{j}\{∞, γ_{j}(∞)\}⟩ = ⟨g, ∑_{j=1}^{m} w \frac{P_{j}(Y, −NX)}{N^{k/2−1}} \{\frac{c}{d} + \frac{i}{d\sqrt{N}} , ∞\}

+ ∑_{j=1}^{m} (P_{j} − w \frac{P_{j}(Y, −NX)}{N^{k/2−1}}) \{\frac{i}{\sqrt{N}} , ∞\}

− ∑_{j=1}^{m} P_{j} \{\frac{b_{j}}{d} + \frac{i}{d\sqrt{N}} , ∞\}⟩ .

Proof. By Proposition 2.1 our condition of transportability implies that we have ∑_{j=1}^{m} P_{j}{∞, γ_{j}(∞)} = ∑_{j=1}^{m} P_{j}{α, γ_{j}(α)}. The steps of the following computation are described below.

⟨g, P_{j}{α, γ_{j}(α)}⟩

= ⟨g, P_{j}\{α, \frac{i}{\sqrt{N}} \} + P_{j}\{\frac{i}{\sqrt{N}} , W (α)\} + P_{j}\{W (α), γ_{j}(α)\}⟩
= ⟨g, w \frac{W (P_{j})}{N^{k/2−1}} \{W (α), \frac{i}{\sqrt{N}} \} + P_{j}\{\frac{i}{\sqrt{N}} , W (α)\} + P_{j}\{W (α), γ_{j}(α)\}⟩
= ⟨g, (w \frac{W (P_{j})}{N^{k/2−1}} − P_{j}) \{W (α), \frac{i}{\sqrt{N}} \} + P_{j}\{W (α), ∞\} − P_{j}\{γ_{j}(α), ∞\}⟩
= ⟨g, w \frac{W (P_{j})}{N^{k/2−1}} \{W (α), ∞\} + (P_{j} − w \frac{W (P_{j})}{N^{k/2−1}} ) \{\frac{i}{\sqrt{N}} , ∞\} − P_{j}\{γ_{j}(α), ∞\}⟩.

In the first step, we break the path from α to γ_{j}(α) into three paths. In the second step, we apply the W -involution to the first term, and use that the action of W is compatible with the pairing ⟨ , ⟩. The third step involves combining the first two terms and breaking up the third. In the final step, we replace \{W (α), i/\sqrt{N}\} by \{W (α), ∞\} + \{∞, i/\sqrt{N}\} and regroup. Taking the sum of both sides of the expression over j from 1 to m gives the first result of the proposition.

Now, following Cremona [1, §2.10.8], in order to simultaneously maximize the imaginary parts of γ_{j}(α) and W (α), we take α = γ_{1}^{−1} (\frac{b_{1}}{d} + \frac{i}{d\sqrt{N}} ). In this

case we have W (α) = \frac{c}{d} + \frac{i}{d\sqrt{N}}, and γ_{j}(α) = \frac{b_{j}}{d} + \frac{i}{d\sqrt{N}}. The second formula then follows.

Remark 3.4. Let γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N). Since the imaginary parts of the terms i/\sqrt{N}, γ_{j}(α) and W (α) in the second part of the proposition are all relatively large, the sums appearing in Equation 6 converge relatively quickly if d is small. However, we emphasize that it is extremely important to choose γ_{j} in Proposition 3.3 with d small; otherwise, the series will converge very slowly.

4 Examples

The example of Section 4.1 illustrates some of the results of this paper for the weight-12 modular form Δ, and Section 4.2 concerns a nonrational form of level 11 and weight 4. The computations below were done using the first author’s implementation of the algorithms of [6] in Magma [3].

4.1 The weight-12 form Δ

Let f = Δ = q · ∏(1 − q^{n})^{24} be the unique normalized eignform in S_{12}(1). The space M_{12}(1; Q) of modular symbols has dimension 3 and is spanned by a_{1} = X^{10}{0, ∞}, a_{2} = X^{8}Y^{2}{0, ∞}, and a_{3} = X^{9}Y {0, ∞}, and the cuspidal subspace S_{12}(1; Q) has dimension 2 and is spanned by a_{2} and a_{3}.

As explained in Example 2.2, there is a transportable modular symbol associated to each nonidentity element γ ∈ SL_{2}(Z). The transportable symbol (2X^{2} + 2XY − Y^{2})^{5}\{∞, \frac{1}{2}\} = −300X^{9}Y \{0, ∞\} is attached to ( \begin{matrix}1 & 1 \\ 2 & 3 \end{matrix} ), and −4665600X^{8}Y^{2}\{0, ∞\} − 87300X^{9}Y \{0, ∞\} is attached to ( \begin{matrix}−8 & 5 \\ 19 & −12 \end{matrix} ). Together these two transportable symbols span S_{12}(1; Q).

The period map Φ_{f} sends X^{i}Y^{10−i}{0, ∞} to 2πi ∫_{0}^{∞} z^{i}f(z) dz. These integrals are, up to scalars, special values of L(f, s) at critical integers, so they could be computed using any of the standard methods. In any case, we obtain an approximation for the period map: Φ_{f}(a_{1}) ∼ 0.0374412812, Φ_{f}(a_{2}) ∼ −0.0159703242, Φ_{f}(a_{3}) ∼ −0.0232962319i. The period lattice Λ of f is spanned by Φ_{f}(\frac{1}{14}a_{2}) and Φ_{f}(\frac{1}{48}a_{3}). (The fractions appear because S_{12}(1; Z) has basis \frac{1}{14}a_{2} and \frac{1}{48}a_{3}.) Since ℂ/Λ is a one-dimensional torus, it makes sense to consider the corresponding elliptic curve over ℂ. This is the elliptic curve y^{2} = x^{3} + c_{4}x + c_{6}, where c_{4} ∼ 28091951348793344.58 and c_{6} ∼ 0. The jinvariant of this curve is approximately 2592849.394270. Is j a transcendental number?

4.2 Level 11, weight 4

The unique normalized eigenform in S_{4}(Γ_{0}(11)) is

f = q + αq^{2} + (−4α + 3)q^{3} + (2α − 6)q^{4} + (8α − 7)q^{5} + ⋯,

where α^{2}−2α−2 = 0. The space M_{4}(Γ_{0}(11); Q) has basis a_{1} = X^{2}{0, ∞}, a_{2} = (64X^{2} + 16XY + Y^{2})\{−\frac{1}{8}, 0\}, a_{3} = (49X^{2} + 14XY + Y^{2})\{−\frac{1}{7}, 0\}, a_{4} = (25X^{2} + 10XY +Y^{2})\{−\frac{1}{5}, 0\}, a_{5} = (100X^{2} +20XY +Y^{2})\{−\frac{1}{10}, 0\}, a_{6} = Y^{2}\{∞, 0\}. The subspace S_{4}(Γ_{0}(11); Q) has basis b_{1} = a_{2} − a_{6}, b_{2} = a_{3} − a_{6}, b_{3} = a_{4} − a_{6}, b_{4} = a_{5} − a_{6}.

As explained in Example 2.2, there is a transportable modular symbol associated to each nonidentity element γ ∈ Γ_{0}(11). For example the transportable symbol (11X^{2} −11XY +Y^{2})\{∞, \frac{10}{11}\} = 11(a_{5} −a_{6}) is associated to γ = ( \begin{matrix}10 & −1 \\ 11 & −1 \end{matrix} ). The symbol −\frac{5}{4}b_{1} + \frac{5}{4}b_{2} − \frac{1}{4}b_{3} + \frac{1}{4}b_{4} is the transportable symbol associated to

( \begin{matrix}5 & −1 \\ 11 & −2 \end{matrix} ). The symbol −\frac{9}{8}b_{1} − \frac{19}{8}b_{2} + \frac{19}{8}b_{3} + \frac{99}{8}b_{4} is associated to ( \begin{matrix}4 & 1 \\ 11 & 3 \end{matrix} ), and

−\frac{27}{8}b_{1} + \frac{11}{8}b_{2} + \frac{9}{8}b_{3} + \frac{49}{8}b_{4} is associated to ( \begin{matrix}3 & −2 \\ 11 & −7 \end{matrix} ). Together these four transportable symbols span S_{4}(Γ_{0}(11); Q).

In order to illustrate Section 2.3, we remark that symbols of the form P {∞, γ(∞)} do not span all of M_{4}(Γ_{0}(11); Q), but they do span a space bigger than S_{4}(Γ_{0}(11); Q). Corollary 2.5 implies that their span contains S_{4}(Γ_{0}(11); Q); however, the symbol Y^{2}\{∞, \frac{1}{11}\} does not lie in S_{4}(Γ_{0}(11); Q).

References

[1] J. E. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997.

[2] , Computing periods of cusp forms and modular elliptic curves, Experiment. Math. 6 (1997), no. 2, 97–107.

[3] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265, Computational algebra and number theory (London, 1993).

[4] J. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66.

[5] L. Merel, Universal Fourier expansions of modular forms, On Artin’s conjecture for odd 2-dimensional representations, Springer, 1994, pp. 59–94.

[6] W. A. Stein, Explicit approaches to modular abelian varieties, Ph.D. thesis, University of California, Berkeley (2000).