Definition of Modular Forms (of Level 1)

Applications of Modular Forms Workshop Goals

Modular Symbols

William Stein

Department of Mathematics University of Washington

July 31, 2006 / MSRI Workshop

Outline

1 Definition of Modular Forms (of Level 1)

2 Applications of Modular Forms

3 Workshop Goals

Recall that

M_{k}(Γ_{1}(N)) = {f : 𝔥 → ℂ such thatf^{[γ]_{k}} = f all γ ∈ Γ_{1}(N), etc.}

Theorem The space M_{k}(Γ_{1}(N)) is computable, i.e., there is an algorithm that takes as input k, N, B and outputs a basis of q-expansions for M_{k}(Γ_{1}(N)) to precision O(q^{B}).

This theorem is not at all obvious from the definition!

The group M_{2} is the free abelian group on symbols {α, β} with

α, β ∈ ℙ^{1}(ℚ) = ℚ ∪ {∞}

modulo the relations

{α, β} + {β, γ} + {γ, α} = 0,

for all α, β, γ ∈ ℙ^{1}(ℚ), and all torsion. More precisely,

M_{2} = (F/R)/(F/R)_{tor},

where F is the free abelian group on all pairs (α, β) and R is the subgroup generated by all elements of the form (α, β) + (β, γ) + (γ, α). Note that M_{2} is a huge free abelian group of countable rank.

Definition of Modular Forms (of Level 1)

Applications of Modular Forms

Workshop Goals

For any integer n ≥ 0, let ℤ[X, Y ]_{n} be the abelian group of homogeneous polynomials of degree n in two variables X, Y .

Remark

Note that ℤ[X, Y ]_{n} is isomorphic to Sym^{n}(ℤ × ℤ) as a group, but certain natural actions are different. In [?], Merel uses the notation ℤ_{n}[X, Y ] for what we denote by ℤ[X, Y ]_{n}.

Now fix an integer k ≥ 2. Set

M_{k} = ℤ[X, Y ]_{k−2} ⊗_{ℤ} M_{2},

which is a torsion-free abelian group whose elements are sums of expressions of the form X^{i}Y^{k−2−i} ⊗ {α, β}. For example,

X^{3} ⊗ {0, 1/2} − 17XY^{2} ⊗ {∞, 1/7} ∈ M_{5}.

Fix a finite index subgroup G of SL_{2}(ℤ). Define a left action of G on ℤ[X, Y ]_{k−2} as follows. If g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ G and P(X, Y ) ∈ ℤ[X, Y ]_{k−2}, let

(gP)(X, Y ) = P(dX − bY , −cX + aY ).

William Stein  Modular Symbols

Definition of Modular Forms (of Level 1)

Applications of Modular Forms

Workshop Goals

Note that if we think of z = (X, Y ) as a column vector, then

(gP)(z) = P(g^{−1}z),

since g^{−1} = ( \begin{matrix}d & −b \\ −c & a \end{matrix}). The reason for the inverse is so that this is a left action instead of a right action, e.g., if g, h ∈ G, then

((gh)P)(z) = P((gh)^{−1}z) = P(h^{−1}g^{−1}z) = (hP)(g^{−1}z) = (g(hP))(z).

Let G act on the left on M_{2} by

g{α, β} = {g(α), g(β)}.

Here G acts via linear fractional transformations, so if g = ( \begin{matrix}a & b \\ c & d \end{matrix} ), then

g(α) = \frac{aα + b}{cα + d} .

For example, useful special cases to remember are that if g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) then

g(0) = \frac{b}{d}  and  g(∞) = \frac{a}{c}.
William Stein  Modular Symbols

Definition of Modular Forms (of Level 1)

Applications of Modular Forms

Workshop Goals

(Here we view ∞ as 1/0 in order to describe the action.) We now combine these two actions to obtain a left action of G on M_{k}, which is given by

g(P ⊗ {α, β}) = (gP) ⊗ {g(α), g(β)}.

For example,

(\begin{matrix}1 & 2 \\ −2 & −3 \end{matrix}) .(X^{3} ⊗ \{0, 1/2\}) = (−3X − 2Y )^{3} ⊗ \{−\frac{2}{3}, −\frac{5}{8}\}

= (−27X^{3} − 54X^{2}Y − 36XY^{2} − 8Y^{3}) ⊗ \{−\frac{2}{3}, −\frac{5}{8}\} .

We will often write P(X, Y ){α, β} for P(X, Y ) ⊗ {α, β}.

Definition (Modular Symbols)

Let k ≥ 2 be an integer and let G be a finite index subgroup of SL_{2}(ℤ). The space M_{k}(G) of weight k modular symbols for G is the quotient of M_{k} by all relations gx − x for x ∈ M_{k}, g ∈ G, and by any torsion.

William Stein  Modular Symbols