Let

L

be the set of lattices in

C

V_{k} = {F : L \to C : F (λ Λ) = λ^{- k} F (Λ) for every λ \in C^{\times}} .

For the normalization used here, the

n

(T_{n} F) (Λ) = n^{k - 1} \sum_{[ Λ ^{'} : Λ ] = n Λ ^{'} \supset Λ} F (Λ^{'}) .

There are finitely many such superlattices, and scaling gives a bijection between those above

Λ

and those above

λ Λ

, so

T_{n} F

again has weight

k

ii

Solution

For

τ \in h

, put

Λ_{τ} = Z τ + Z

f_{F} (τ) = F (Λ_{τ}) .

γ = (a c b d) \in Γ (1)

, then

Λ_{γ τ} = (c τ + d)^{- 1} Λ_{τ},

f_{F} (γ τ) = (c τ + d)^{k} f_{F} (τ)

. Conversely, if

f

has this invariance and

Λ = Z ω_{1} + Z ω_{2}

with

Im (ω_{1} / ω_{2}) > 0

F_{f} (Λ) = ω_{2}^{- k} f (ω_{1} / ω_{2}) .

The modular transformation law makes this independent of the oriented basis. These constructions are inverse, identifying

V_{k}

with

W_{k}

Transporting

T_{n}

through this identification gives

(T_{n} f) (τ) = n^{k - 1} \sum_{d > 0 a d = n} d^{- k} \sum_{b mod d} f (\frac{a τ + b}{d}) .

This defines the Hecke operator on

W_{k}

There are no discussions about this article yet.

There are currently no matching articles.