Computing Period Lattices of Newforms (Preliminary)

William Stein and Helena Verrill

April, 1999

Abstract We give an algorithm to numerically approximate the period lattice and real volume of any modular newforms f ∈ S_{k}(Γ, C) of positive even weight. The computation of special values of the associated L-functions is also discussed. Numerical examples are provided.

1 Modular Symbols

Let Γ = Γ_{0}(N) or Γ_{1}(N), and let k be an even positive integer. Let M be the abelian group generated by symbols {α, β} with α, β ∈ P^{1}(Q) modulo the relations

{α, β} + {β, γ} + {γ, α} = 0

and modulo any torsion.

[figure: α; β; γ]

Let

V_{k−2} = Sym^{k−2}(ZX + ZY)

be the Z-module of homogeneous polynomials in X and Y of degree k − 2. Define a left action of g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ on the tensor product V_{k−2} ⊗ M by

g(P (X, Y) ⊗ {α, β}) = P (dX − bY, −cX + aY) ⊗ {g(α), g(β)}.

The space of modular symbols of weight k for Γ is the largest quotient of V_{k−2} ⊗ M on which Γ acts trivially. If we let I = {1 − g : g ∈ Γ} ⊂ Z[Γ] then

M_{k}(Γ) = \frac{V_{k−2} ⊗ M}{I(V_{k−2} ⊗ M)} = H_{0}(Γ, V_{k−2} ⊗ M).

The cuspidal modular symbols are S_{k}(Γ) = ker(δ), where

δ : M_{k}(Γ) → ⨁_{[α]∈Γ\P^{1}(Q)} Z[α]

is given by

δ(P (X, Y) ⊗ {α, β}) =?.

The Eichler-Shimura theory gives an isomorphism

S_{k}(Γ, C) ⊕ \overline{S}_{k}(Γ, C) ⊕ Eis(Γ, C) ≅ H^{1}(Γ, V_{k−2}^{∗} ⊗ C)

For example, when k = 2 the map

S_{2}(Γ, C) → H^{1}(Γ, C) = Hom(Γ, C)

send f(z) to the map γ ↦ ∫_{0}^{γ(0)} f(z)dz.

The cup product pairing

H_{0}(Γ, V_{k−2} ⊗ M) × H^{1}(Γ, V_{k−2}^{∗} ⊗ C) → H^{0}(Γ, C) = C

gives rise to the period map we will study.

2 Computing Periods

Let Γ = Γ_{0}(N) or Γ_{1}(N). In this section we describe how to numerically approximate the period lattice associated to a cuspidal newform

f = ∑ a_{n}q^{n} ∈ S_{k}(Γ, C), k ≥ 2.

We do not require that the Fourier coefficients a_{n} of f lie in Q, nor that the weight equals 2. In the special case k = 2 and all a_{n} ∈ Q, see [CR]. The algorithm given only applies when k is even, though we expect that there is a generalization to odd k.

Let T = Z[T_{1}, T_{2}, T_{3}, …] denote the Hecke algebra, equipped with its action on modular forms and modular symbols for Γ. Set

𝔭_{f} = Ann_{T}(f).

Let f^{(1)}, …, f^{(d)} be the Gal(\overline{Q}/Q) conjugates of f. Let M_{k}(Γ, Q) be the space of modular symbols of weight k for Γ (tensor Q) and S_{k}(Γ, Z) ⊂ M_{k}(Γ, Q) the subspace of integral cuspidal symbols. There is a pairing

⟨ , ⟩ : M_{k}(Γ, Q) × S_{k}(Γ, C) → C
⟨P (X, Y){α, β}, f⟩ = 2πi ∫_{α}^{β} f(z)P (z, 1)dz.

The period map associated to f is

Φ_{f} : M_{k}(Γ, Q) → C^{d}
Φ_{f}(x) = (⟨x, f^{(1)}⟩, …, ⟨x, f^{(d)}⟩).

Let

Λ_{f} = Φ_{f}(S_{k}(Γ, Z)) ⊂ C^{d}.

The following is an algorithm to numerically approximate Φ_{f} and Λ_{f}.

1. [Compute left eigenspace.] Compute a Q-basis φ_{1}, …, φ_{2d} for

Hom(M_{k}(Γ, Q), Q)[𝔭_{f}].

2. [Dual basis.] Let w_{1}, …, w_{2d} be the Q-basis for M_{k}(Γ, Q)[𝔭_{f}] dual to φ_{1}, …, φ_{2d}, so φ_{i}(w_{j}) = δ_{ij}. It is not necessary to compute the w_{i} though this could be done by inverting the matrix whose rows are the φ_{i}. Observe that if v ∈ M_{k}(Γ, Q) then

⟨v, f⟩ = ∑_{i=1}^{2d} φ_{i}(v)⟨w_{i}, f⟩.

3. [Compute Z-basis for Λ_{f}.] Compute a Z-basis z_{1}, …, z_{2d} for the

lattice

{(φ_{1}(x), …, φ_{2d}(x)) : x ∈ S_{k}(Γ, Z)} ⊂ Q^{2d}.

4. [Find good periods.] Locate v_{1}, … v_{2d} ∈ M_{k}(Γ, Q) so that

• Each v_{i} is of the form

P_{g}(X, Y){0, g(0)}

with g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ, c > 0, and

P_{g}(X, Y) = ((1 − ad)X^{2} − b(d − a)XY + b^{2}Y^{2})^{\frac{k−2}{2}} .

Note that P is an eigenvector for the action of g on V_{k−2} with eigenvalue 1. Remarks:

(a) This is the step which only makes sense for even weight. We have not yet found a replacement for odd weight.

(b) It is still necessary to prove the existence of enough g’s. This we have not done. In practice a few randomly choosen g’s seem to do the job.

• The matrix (φ_{i}(v_{j})) is invertible.

5. [Compute approximate periods.] The form of each v_{i} insures that we can obtain good numerical approximations to ⟨v_{i}, f⟩. Write

v_{i} = P_{g}(X, Y)\{0, g(0)\} with g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and c > 0. We have ⟨v_{i}, f⟩ =

∑_{n≥1} a_{n}c_{n}, where

c_{n} = 2πi ∫_{0}^{g(0)} e^{2πinz}P_{g}(z, 1)dz.

To compute c_{n}, let y_{0} = 1/c, x_{1} = −d/c, x_{2} = a/c. Then

c_{n} = 2πi ∫_{x_{1}+iy_{0}}^{x_{2}+iy_{0}} e^{2πinz}P_{g}(z, 1)dz.

Proof. Let α = x_{1} + iy_{0} and β = x_{2} + iy_{0}. Observe that g(α) = β and (key fact) P_{g}(aX + bY, cX + dY) = P_{g}(X, Y), so

g^{−1}(P_{g}(X, Y){x, y}) = P_{g}(X, Y){g^{−1}(x), g^{−1}(y)}

for any x and y. We thus have

0 = P_{g}(X, Y)({0, g(0)} + {g(0), g(α)} + {g(α), 0})
= P_{g}(X, Y){0, g(0)} + P_{g}(X, Y){0, α} + P_{g}(X, Y){g(α), 0}
= P_{g}(X, Y)({0, g(0)} − {α, g(α)})

so P_{g}(X, Y){0, g(0)} = P_{g}(X, Y){α, g(α)}.

The following formula is useful in evaluating c_{n}.

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

Since y_{0} > 0, the c_{n} → 0 quickly.

Proof. Repeated integration by parts.

6. [Solve for periods of dual basis.] We have a matrix equation

(\begin{matrix}φ_{1}(v_{1}) & ⋯ & φ_{1}(v_{2d}) \\ ⋯ & ⋯ & ⋯ \\ φ_{2d}(v_{1}) & ⋯ & φ_{2d}(v_{2d}) \end{matrix}) · (\begin{matrix}⟨w_{1}, f⟩ \\ … \\ ⟨w_{2d}, f⟩ \end{matrix}) = (\begin{matrix}⟨v_{1}, f⟩ \\ … \\ ⟨v_{2d}, f⟩ \end{matrix}) .

Inverting we obtain numerical approximations α_{i} ∼ ⟨w_{i}, f⟩. Compute the approximations ⟨v_{i}, f^{(j)}⟩ for each of the Galois conjugates f^{(j)} of f in order to obtain approximations

α_{i}^{(j)} ∼ ⟨w_{i}, f^{(j)}⟩ ∈ C,

for 1 ≤ i ≤ 2d and 1 ≤ j ≤ d.

7. [Compute period map.] The period map is approximated as

Φ_{f}(v) ∼ (\begin{matrix}α_{1}^{(1)} & ⋯ & α_{2d}^{(1)} \\ ⋯ & ⋯ & ⋯ \\ α_{1}^{(d)} & ⋯ & α_{2d}^{(d)} \end{matrix}) · (\begin{matrix}φ_{1}(v) \\ | \\ φ_{2d}(v) \end{matrix})

8. [Compute period lattice.] The (approximate) period lattice Λ_{f} has as basis the columns of the following product:

(\begin{matrix}α_{1}^{(1)} & ⋯ & α_{2d}^{(1)} \\ ⋯ & ⋯ & ⋯ \\ α_{1}^{(d)} & ⋯ & α_{2d}^{(d)} \end{matrix}) · (\begin{matrix}| & ⋯ & | \\ z_{1} & ⋯ & z_{2d} \\ | & ⋯ & | \end{matrix})

2.1 Special Values

Let

L(f, s) = ∫_{0}^{∞} f(z)z^{s−1}dz = ∑ a_{n}n^{−s}

be the Mellin transform of f. We can approximate the special value L(A_{f}, 1) = ∏_{i} L(f^{(i)}, 1) by observering that

Φ_{f}(Y^{k−2}{0, ∞}) = (L(f^{(1)}, 1), …, L(f^{(d)}, 1)).

2.2 The Real Period

The complex torus A_{f} = C^{d}/Λ_{f} is of interest. When k = 2 it has the structure of abelian variety with a good integral model, it is naturally a quotient of Jac(X_{Γ}).

Assume for the rest of this section that the Fourier coefficients of f are totally real. Then the lattice Λ_{f} is invariant under complex conjugation. Thus we may consider the real points A_{f}(R) = (C^{d}/Λ_{f})^{+} of the complex torus A_{f}. Let Ω_{f} := Vol(A_{f}(R)) denote the volume of the identity component of A_{f}(R) with respect to the measure on A_{f} induced by the standard Lebesgue measure on C^{d}. Then

Ω_{f} = Vol(R^{d}/Λ_{f}^{+}) · c_{∞}

where

c_{∞} = #(A_{f}(R)/A_{f}(R)^{0})

is the number of connected components of A_{f}(R).

1. [Compute conjugation on Z = Zz_{1} + ⋯ + Zz_{2d}] The action of complex conjugation on Λ_{f} can be computed as follows. Note that conjugation commutes with the period map Φ_{f}. If z_{i} lifts to x ∈ S_{k}(Γ, Z), then

z_{i}^{∗} = (φ_{1}(x^{∗}), …, φ_{2d}(x^{∗})).

Compute the matrix C representing conjugation with respect to the basis z_{1}, …, z_{2d}. Compute a Z-basis b_{1}, …, b_{d} for Z^{+} = Ker(C − I).

2. [Compute Vol(A_{f}(R)^{0}) = Vol(R^{d}/Λ_{f}^{+})]

Vol(R^{d}/Λ_{f}^{+}) ∼ det \{(\begin{matrix}α_{1}^{(1)} & ⋯ & α_{2d}^{(1)} \\ ⋯ & ⋯ & ⋯ \\ α_{1}^{(d)} & ⋯ & α_{2d}^{(d)} \end{matrix}) · (\begin{matrix}| & ⋯ & | \\ b_{1} & ⋯ & b_{d} \\ | & ⋯ & | \end{matrix})\}

Order the b_{i} so that det > 0.

3. [Compute c_{∞}] Let \overline{C} be the the map induced by complex conjugation on Λ_{f}/2Λ_{f} = Λ_{f} ⊗ F_{2}. Then

c_{∞} = 2^{dim(ker(\overline{C}−1))−d}.

Proof. We must compute the order of the component group

Ψ = \frac{A_{f}(R)}{A_{f}(R)^{0}} = \frac{(C^{d}/Λ_{f})^{+}}{R^{d}/Λ_{f}^{+}} .

For v ∈ C^{d} denote by \overline{v} its complex conjugate and by [v] its image in C^{d}/Λ_{f}. If [v] ∈ (C^{d}/Λ_{f})^{+} then [v] = [\overline{v}] so since v + \overline{v} ∈ R^{d} we have

2[v] = [v] + [\overline{v}] ∈ R^{d}/Λ_{f}^{+},

so Ψ is annihilated by 2. Thus there is λ ∈ Λ_{f} so that 2v + λ ∈ R^{d}, and so v + \frac{1}{2}λ ∈ R^{d}, i.e., v can be written as something in \frac{1}{2}Λ_{f} plus something in R^{d}. This means that Ψ is generated by the image of (\frac{1}{2}Λ_{f}/Λ_{f})^{+}. Thus

Ψ ≅ \frac{(\frac{1}{2}Λ_{f}/Λ_{f})^{+}}{(\frac{1}{2}Λ_{f} ∩ R^{d})/Λ_{f}^{+}} ≅ \frac{(Λ_{f}/2Λ_{f})^{+}}{Λ_{f}^{+}/2Λ_{f}^{+}}

Consequently

dim_{F_{2}} Ψ = dim(Λ_{f}/2Λ_{f})^{+} − dim Λ_{f}^{+}/2Λ_{f}^{+} = dim(ker(\overline{C} − 1)) − d.

2.3 The Ratio L(A_{f}, 1)/Ω_{f}

I conjecture that the following is true:

Let F (x) = (φ_{1}(x), …, φ_{2d}(x)). Let C be the map on the image of F induced by the ∗ involution on S_{k}(Γ, Z) and \overline{C} its mod 2 reduction. Then

|\frac{L(A_{f}, 1)}{Ω_{f}} | = [F (S_{k}(Γ, Z))^{+} : F (Te)] · 2^{dim(ker(\overline{C}−1))−d}

where [A : B] is the absolute value of the determinant of a change of basis matrix sending A to B.

3 Examples

4 Tables

References

[CR] J.E. Cremona, Algorithms for modular elliptic curves, 2nd edition, Cambridge University Press, (1997).