Lattices of modular symbols for Γ_{0}(N), weight k ≥ 2

Helena Verrill

Hannover University

Notes for a talk given at Harvard, October 3, 2001, on work with William Stein.

We will consider 3 lattices

[figure: S_{k}(Γ_{0}(N), Z); H_{k−1}(W, Z); T _{k}(Γ_{0}(N), Z)]

S_{k}(Γ_{0}(N), Z) = cuspidal modular symbols of weight k, level N (well known)
T _{k}(Γ_{0}(N), Z) = transportable modular symbols of weight k, level N (new; Stein + —)
H_{k−1}(W, Z) = k − 2 cycles in W

Where W is the k−2 fold fibre product of the universal family E of elliptic curves with level N structure, over X_{0}(N).

\xymatrix{E \ar[d] & E_{t} \ar@{|->}[d] & W \ar[d] & \overbrace{E_{t} × ⋯ × E_{t}}^{k−1} \ar@{|->}[d] \\ X_{0}(N) & t & X_{0}(N) & t}

How are these lattices related?

Next: motivation

definitions algorithms

1 Motivation

Reasons T _{k}(Γ_{0}(N), Z) might be a better lattice than S_{k}(Γ_{0}(N), Z):

• S_{k}(Γ_{0}(N), Z) was defined by Manin/Shokurov/Merel, and allows one to explicitly compute with modular forms. However, H_{k−1}(W, Z) is a more natural geometrical object. T _{k}(Γ_{0}(N), Z) is contained in both.

• T _{k}(Γ_{0}(N), Z) is useful for computing periods,

• and essential for computing the intersection pairing on S_{k}(Γ_{0}(N), Z)

• Let f be a cuspidal Hecke eigenform f, with Annihilator I_{f} = Ann(f) ⊂ T, where T is the Hecke Algebra, then computations suggest that

\frac{modular degree of f}{[S_{k}(Γ, Z)[I_{f}] : T _{k}(Γ, Z)[I_{f}]] × det (intersection pairing on S_{k}(Γ, Z)[I_{f}])}

is in Z, divisible only by primes dividing (k − 2)!. Perhaps the intersection pairing (below) is wrong by a scale factor, since W is not quite the right object.

2 Definitions of S_{k}(Γ, Z) [Manin/Shokurov/Merel]

Notation Γ = Γ_{0}(N). H will denote either 𝔥^{∗}, the upper half plane union the cusps, or P^{1}(Q). V_{k} = homogeneous polynomials of degree k with coefficients in Z.

Γ acts linearly on V_{k}: g(P (X, Y)) = P (g^{−1}(X, Y)), and by fractional linear transformation on H.

Boundary symbols  geometrical picture:

B = ⟨{α}|α ∈ H⟩ B_{k} = V_{k−2} ⊗ B = ⟨{P {α}|α ∈ H}⟩ B_{k}(Γ, Z) = B_{k}/⟨b − gb|b ∈ B_{k}⟩/torsion

think of as k − 2 cycles in fibres of W

Modular symbols  geometrical picture:

M = ⟨{α, β}|α, β ∈ H⟩ ⊂ B where {α, β} = {β} − {α} M_{k} = V_{k−2} ⊗ M = ⟨{P {α, β}|P ∈ V_{k−2}α ∈ H}⟩ M_{k}(Γ, Z) = M_{k}/⟨m − gm|m ∈ M_{k}⟩/torsion think of as k−1 cycles in W, with boundary

Cuspidal symbols  geometrical picture:
S_{k}(Γ, Z) is the kernel of the boundary map  think of as closed k − 1 cycles in W
\xymatrix{0 \ar[r] & S_{k}(Γ, Z) \ar[r] & M_{k}(Γ, Z) \ar[r]^{δ} & B_{k}(Γ, Z) \ar[r] & 0,}

where δ(P {α, β}) = P {β} − P {α}. For R = Q or C, let S_{k}(Γ, R) = S_{k}(Γ, Z) ⊗ R, etc. Note S_{k}(Γ, Z) is the same whether we take H = 𝔥^{∗} or H = P^{1}(Q). Using P^{2}(Q) is best for finding S_{k}(Γ, Z) as a Hecke module; using 𝔥^{∗} is better for computing periods.

Theorem 2.1 (Manin/Shokurov). Let Γ = Γ_{0}(N), and let S_{k}(Γ, C) be the space of cusp forms, and \overline{S_{k}(Γ, C)}, the antiholomorphic cusp forms of weight k, level N. There is a perfect pairing

(S_{k}(Γ, C) ⊕ \overline{S_{k}(Γ, C)} × S_{k}(Γ, C) → C

⟨f + g, P (X, Y)\{α, β\}⟩ = ∫_{α}^{β} P (τ, 1)f(τ)dτ + ∫_{α}^{β} P (\overline{τ}, 1)g(τ)d\overline{τ},

which induces an Hecke invariant isomorphism between S_{k}(Γ, C) and the dual of S_{k}(Γ, C) ⊕ \overline{S_{k}(Γ, C)}.

This means that to compute the eigenvalues of T on S_{k}(Γ, C), we just need to compute the eigenvalues on S_{k}(Γ, C).

3 Transportable symbols [Stein/—]

Use integration by parts to show that:

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

where

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

Problem: if α ∈ Q then this sum does not converge well. Weight 2 solution:[Manin] There is a surjective homomorphism

Γ → S_{2}(Γ, Z)
g ↦ {0, g0}

Fact: {0, g0} = {α, gα} for all α ∈ 𝔥^{∗}. Proof: {0, g0} = {0, α} + {α, gα} + gα, g0}, and {0, α} = {g0, gα} = −{gα, g0}. So, we can “transport” weight 2 symbols away from Q, and find period integrals.

3.1 Transportable symbols for k > 2 (arbitrary character)

Definition 3.1. A modular symbol is transportable if it can be written as

∑ P_{i}{0, g_{i}0} with ∑ P_{i} = ∑ g^{−1}P_{i}.

Lemma 3.2. For a transportable symbol as above,

∑ P_{i}{0, g_{i}0} = ∑ P_{i}{α, g_{i}α}.

for all α ∈ 𝔥^{∗}

Definition 3.3.

T _{k}(Γ, Z) = ⟨∑ P_{i}{0, g_{i}0}| ∑ P_{i} = ∑ g^{−1}P_{i}⟩ ⊂ S_{k}(Γ, Z)

Theorem 3.4. T _{k}(Γ, Z) has finite index in S_{k}(Γ, Z).

Question: What is the index [T _{k}(Γ, Z) : S_{k}(Γ, Z)] ?

Lemma 3.5. T _{k}(Γ, Z) is spanned by v_{j} = ∑_{i=1}^{n} P_{i,j}{0, g_{i}0} where g_{1}, g_{2}, … g_{n} is a set of generators for Γ, and (P_{1,j}, … P_{n,j}), with j = 1, … (k − 1)(n − 1), is a basis for the kernel of ⊕(1 − g_{i}^{−1}) : ⊕_{i}^{n}V_{k−2} → V_{k−2}.

Conjecture 3.6. [S_{4} : T _{4}] is divisible only by 2 and primes dividing N.

4 Intersection pairing

On H_{k−1}(W, Z) there is an intersection pairing, which is unimodular, antisymmetric, and Hecke invariant. This gives a pairing on S_{k}(Γ_{0}(N), Z), satisfying

⟨a, b⟩ = −⟨b, a⟩ and ⟨T a, b⟩ = ⟨a, T b⟩ ∀T ∈ T.

Computing the intersection pairing allows us to find the index

[S_{k}(Γ_{0}(N), Z) : H_{k−1}(W, Z)].

4.1 Algorithm for computing the intersection pairing

Idea: Generalize Merel’s method to higher weight: Claim: We can write symbols in terms of images of P {0, ∞} and of Q{ρ, ρ^{2}}, where ρ = \frac{1}{2}(1+\sqrt{3}). Then we can compute the intersection of h(P \{0, ∞\}) with g(Q{ρ, ρ^{2}}), for g, h ∈ SL_{2}(Z). (Even though this is not well defined—need to take sums of these symbols to get elements of S_{k}(Γ_{0}(N), Z).) proof of claim

Manin: All modular symbols can be written as ∑ P_{i}{g_{i}0, g_{i}∞}.

They can also all be written as ∑ P_{i}{ρ, g_{i}ρ} with ∑ P_{i} = ∑ g^{−1}P_{i}.

Suppose g = ∏_{i=1}^{n} W_{i}^{ε_{i}} for W_{i} ∈ \{T = ( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ) , S = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )\}, ε_{i} ∈ \{1, −1\}

For simplicity take all ε = 1. Let h_{m} = ∏_{i=1}^{m−1} W_{i} Then

P {ρ, gρ} = P {ρ, W_{1}W_{2} … W_{n}ρ}
= P ({ρ, W_{1} … W_{n−1}ρ} + {W_{1} … W_{n−1}ρ, W_{1} … W_{n}ρ})
= P {ρ, W_{1} … W_{n−1}ρ} + (h_{n}^{−1}P){ρ, W_{n}ρ}
= ∑ h_{m}^{−1}(P {ρ, W_{m}ρ}) = ∑ h_{m}^{−1}(P {ρ, ρ^{2}})

Heuristic argument for computing the intersection pairing

P g{ρ, ρ^{2}} corresponds to a k − 1 cycle in W over the image of a path from ρ to ρ^{2}, and u(Q{0, ∞}) corresponds to a k − 1 cycle in W over the image of a path from ρ to ρ^{2}.

The corresponding paths in X_{0}(N) intersect ⟺ gu^{−1} ∈ Γ or gSu^{−1} ∈ Γ

If gu^{−1} ∈ Γ then ⟨P g{ρ, ρ^{2}}, uQu{0, ∞}⟩ = ⟨P, gQ⟩.
If gSu^{−1} ∈ Γ then ⟨P g{ρ, ρ^{2}}, uQu{0, ∞}⟩ = −⟨P, gSQ⟩.

Here, ⟨P, R⟩ for R = gQ or gSQ, is given by finding the intersection pairing of corresponding k −2 cycles in the fibre over the intersection point of the paths g{ρ, ρ^{2}} and u{0, ∞} in X_{0}(N).

The fibre is given by a product of k − 2 copies of an elliptic curve. The intersection pairing on E × ⋯ × E can be given by a matrix with alternating +1 and −1 on the anti diagonal. We “lift” polynomials in a unique way to an invariant subspace of H_{k−2}(W_{t}, Z), and find that the intersection pairing is given by

⟨X^{m}Y^{k−2−m}, X^{k−2−m}Y^{m}⟩ = \frac{(−1)^{m}}{(\begin{matrix}k − 2 \\ m \end{matrix})}.