Hecke Operators on Modular Symbols

Hecke Operators on Manin Symbols

Using Hecke Operators to Compute Modular Forms

Hecke Operators and q-Expansions

William Stein

Department of Mathematics University of Washington

August 3, 2006 / MSRI Workshop

Outline

1 Hecke Operators on Modular Symbols

2 Hecke Operators on Manin Symbols

3 Using Hecke Operators to Compute Modular Forms

Hecke Operators on Modular Symbols Hecke Operators on Manin Symbols

Using Hecke Operators to Compute Modular Forms

Hecke Operators on Modular Symbols

Let Γ = Γ_{1}(N) or Γ_{0}(N). For any positive integer n, let

X_{n} = \{(\begin{matrix}a & b \\ 0 & d \end{matrix}) ∈ Mat_{2}(ℤ) : a ≥ 1, ad = n, and 0 ≤ b < d\} .

Note that the set X_{n} is in bijection with the set of subgroups of ℤ^{2} of index n, where ( \begin{matrix}a & b \\ c & d \end{matrix} ) corresponds to L = ℤ · (a, b) + ℤ · (0, d), as one can see using Hermite normal form. If f is a modular form of weight k, then

T_{n}(f ) = ∑_{g∈X_{n}} f^{[γ]_{k}}.

For a modular symbol P{α, β} we define

T_{n}(x) = ∑_{g∈X_{n}} g(P{α, β}).

We have

⟨T_{n}(f ), x⟩ = ⟨f , T_{n}(x)⟩.

Hecke Operators on Modular Symbols Hecke Operators on Manin Symbols

Using Hecke Operators to Compute Modular Forms

Example: Hecke Operators on Modular Symbols

When k = 2 and p is a prime not dividing N, we have

T_{p}(\{α, β\}) = (\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) \{α, β\} + ∑_{r mod p}(\begin{matrix}1 & r \\ 0 & p \end{matrix}) \{α, β\}.

Example For example, when N = 11 we have

T_{2}{0, 1/5} = {0, 2/5} + {0, 1/10} + {1/2, 3/5}
= −2{0, 1/5}.

Hecke Operators on Modular Symbols Hecke Operators on Manin Symbols

Using Hecke Operators to Compute Modular Forms

Hecke Operators on Manin Symbols

If S is a subset of GL_{2}(ℚ), let

S̃ = {g̃ : g ∈ S},

where

g̃ = (\begin{matrix}d & −b \\ −c & a \end{matrix}) = det(g) · g^{−1}.

Also, for any ring R and any subset S ⊂ Mat_{2}(ℤ), let R[S] denote the free R-module with basis the elements of S, so the elements of R[S] are the finite R-linear combinations of the elements of S. An element

h = ∑ u_{M}[M] ∈ ℂ[Mat_{2}(ℤ)_{n}]

satisfies condition C_{n} if for every K ∈ Mat_{2}(ℤ)_{n}/ SL_{2}(ℤ), we have that

∑_{M∈K}u_{M}([M∞] − [M0]) = [∞] − [0] ∈ ℂ[P^{1}(ℚ)].  (2.1)

Suppose h satisfies condition C_{n}.

Theorem (Manin)

For any Manin symbol [P, g] ∈ M_{k}(Γ),

T_{n}([P, (u, v)]) = ∑_{M}u_{M}[P(aX + bY , cX + dY ), (u, v)M]. (2.2)

Here (u, v) corresponds to a coset of Γ in SL_{2}(ℤ), and if (u′, v′) = (u, v)M ∈ (ℤ/Nℤ)^{2}, and gcd(u′, v′, N) ≠ 1, then we omit the corresponding summand.

Hecke Operators on Modular Symbols Hecke Operators on Manin Symbols

Using Hecke Operators to Compute Modular Forms

Heilbronn Matrices

Proposition (Merel)

The element

∑_{a>b≥0 ; d>c≥0 ; ad−bc=n} [(\begin{matrix}a & b \\ c & d \end{matrix})] ∈ ℤ[Mat_{2}(ℤ)_{n}]

satisfies condition C_{n}.

Merel’s proof is not too difficult, but takes two pages. He also gives several other examples of elements that satisfy condition C_{n}. They each have ∼ n log(n) terms, which has implications for complexity of computing Hecke operators.

Example: Computing a Hecke Operator

Hecke Operators on Modular Symbols Hecke Operators on Manin Symbols

Using Hecke Operators to Compute Modular Forms

Linear Functionals

Gabor explained that there is an isomorphism of vector spaces

Ψ : S_{k}(Γ) ⟶^{≅} Hom(𝕋′_{ℂ}, ℂ)  (3.1)

that sends f ∈ S_{k}(Γ) to the homomorphism

t ↦ a_{1}(t(f )).

For any ℂ-linear map φ : 𝕋′_{ℂ} → ℂ, let

f_{φ} = ∑_{n=1}^{∞}φ(T_{n})q^{n} ∈ ℂ[[q]].

Lemma

The series f_{φ} is the q-expansion of Ψ^{−1}(φ) ∈ S_{2}(Γ).

Proof.

Note that it is not even a priori obvious that f_{φ} is the q-expansion of a modular form. Let g = Ψ^{−1}(φ), which is by definition the unique element of S_{2}(Γ) such that ⟨g, T_{n}⟩ = φ(T_{n}) for all n. By Lemma ?? , we have

⟨f_{φ}, T_{n}⟩ = a_{1}(T_{n}(f_{φ})) = a_{n}(f_{φ}) = φ(T_{n}),

so ⟨f_{φ} − g, T_{n}⟩ = 0 for all n. Proposition ?? implies that f_{φ} − g = 0, so f_{φ} = g = Ψ^{−1}(φ), as claimed.

Conclusion: The cusp forms f_{φ}, as φ varies through a basis of Hom(𝕋′_{ℂ}, ℂ), form a basis for S_{2}(Γ). In particular:

We can compute S_{2}(Γ) by computing Hom(𝕋′_{ℂ}, ℂ), where we compute 𝕋′ in any way we want, e.g., using a space that contains an isomorphic copy of S_{2}(Γ).

William Stein  Hecke Operators and q-Expansions

Hecke Operators on Modular Symbols Hecke Operators on Manin Symbols

Using Hecke Operators to Compute Modular Forms

Example: S_{2}(Γ_{0}(11))

The smallest N with S_{2}(Γ_{0}(N)) ≠ 0 is N = 11.

sage: M = ModularSymbols(11); M.basis()
((1,0), (1,8), (1,9))
sage: S = M.cuspidal_submodule(); S
Dimension 2 subspace of a modular symbols space of level 11

We compute a few Hecke operators, then read off a nonzero cusp form, which forms a basis for S_{2}(Γ_{0}(11)):

sage: S.T(2).matrix()
[-2  0]
[ 0 -2]
sage: S.T(3).matrix()
[-1  0]
[ 0 -1]

Thus

f_{0,0} = q − 2q^{2} − q^{3} + ⋯ ∈ S_{2}(Γ_{0}(11))

forms a basis for S_{2}(Γ_{0}(11)).

Hecke Operators on Modular Symbols Hecke Operators on Manin Symbols

Using Hecke Operators to Compute Modular Forms

Example: S_{2}(Γ_{0}(33))

We compute a basis for S_{2}(Γ_{0}(33)) to precision O(q^{6}).

sage: M = ModularSymbols(33)
sage: S = M.cuspidal_submodule(); S
Dimension 6 subspace of a modular symbols space of level 33

Thus dimS_{2}(Γ_{0}(33)) = 3.

sage: R.<q> = PowerSeriesRing(QQ)
sage: a = [S.T(n).matrix()[0,0] for n in range(6)]
sage: f00 = sum(v[n]*q^n for n in range(6)) + O(q^6)
sage: f00
q - q^2 - q^3 + q^4 + O(q^6)

This gives us one basis element of S_{2}(Γ_{0}(33)). It remains to find two others. We find

sage: a = [S.T(n).matrix()[0,1] for n in range(6)]
sage: f01 = sum(v[n]*q^n for n in range(1,6)) + O(q^6)
sage: f01
-2*q^3 + O(q^6)

and

sage: a = [S.T(n).matrix()[1,0] for n in range(6)]
sage: f10 = sum(v[n]*q^n for n in range(1,6)) + O(q^6)
sage: f10
q^3 + O(q^6)

This third one is (to our precision) a scalar multiple of the second, so we look further.

sage: a = [S.T(n).matrix()[1,1] for n in range(6)]
sage: f11 = sum(v[n]*q^n for n in range(1,6)) + O(q^6)
sage: f11
q - 2*q^2 + 2*q^4 + q^5 + O(q^6)

This latter form is clearly not in the span of the first two. Thus we have the following basis for S_{2}(Γ_{0}(33)) (to precision O(q^{6})):

f_{00} = q − q^{2} − q^{3} + q^{4} + ⋯
f_{11} = q − 2q^{2} + 2q^{4} + q^{5} + ⋯
f_{10} = q^{3} + ⋯

William Stein  Hecke Operators and q-Expansions

Hecke Operators on Modular Symbols Hecke Operators on Manin Symbols

Using Hecke Operators to Compute Modular Forms

Eigenforms

1 Use the Atkin-Lehner theory of newforms to write S_{k}(N, ε) in terms of

new subspaces of S_{k}(M, ε) with M | N. 2 Compute a basis for the new subspace of V = S_{k}(M, ε) by decomposing

V as a direct sum of simple 𝕋-modules (using linear algebra). 3 Suppose W ⊂ V is a simple module and fix x ∈ W nonzero. 4 Define a surjective map ϕ : 𝕋_{ℚ} → W given by t ↦ tx. 5 This endows W with an algebra structure (in which x corresponds to 1).

In particular, you can choose and fix an isomorphism W → K = ℚ(a_{2}, a_{3}, …), given in terms of a power basis.

6 Store either the T_{p}(x) or T_{n}(x), depending on the intended application. 7 Note – one can recover a_{n} for n composite from the a_{p} for p prime via

arithmetic...