Hecke algebra

Let $f$ be a modular form for $\mathrm{\Gamma }$ a congruence subgroup of ${\text{SL}}_2(\mathbb{Z})$.

$$f(z)=\sum _{n=0}^{\infty }{a}_n{q}^n$$ (1)

where $q=e^{2i\pi \tau }$.

For $m\in \mathbb{N}$, let $T_{m}f(z)=\sum _{n=0}^{\infty }{b}_n{q}^n$ with :

$$b_n=\sum _{d|\gcd (m,n)}^{}{d}^{k-1}{a}_{mn/d^2}$$ (2)

In particular, for $p$ a prime, $T_{p}f(z)=\sum _{n=0}^{\infty }{b}_n{q}^n$ with;

$$b_n=a_{pn}+p^{k-1}{a}_{n/p}$$ (3)

where $a_{n/p}=0$ if $n$ is not divisible by $p$.

The operator $T_n$ is a linear operator on the space of modular forms called a Hecke operator.

The Hecke operators leave the space of modular forms and cusp forms invariant and turn out to be self-adjoint for a scalar product called the Petersson scalar product. In particular they have real eigenvalues. Hecke operators also satisfy multiplicative properties that are best summarized by the formal identity:

$$\sum _{n=1}^{\infty }{T}_n{n}^{-s}=\prod _{𝑝}{(1-T_p{p}^{-s}+p^{k-1-2s})}^{-1}$$ (4)

That equation in particular implies that $T_m{T}_n=T_n{T}_m$ whenever $\gcd (n,m)=1$.

The set of all Hecke operators is usually denoted $𝕋$ and is called the Hecke algebra.

0.1 Group algebra example

Definition 0.1 Let $G_{lcd}$ be a locally compact totally disconnected group; then the Hecke algebra $\mathrm{H}\mathit{}\mathrm{(}{G}_{l\mathit{}c\mathit{}d}\mathrm{)}$ of the group $G_{lcd}$ is defined as the convolution algebra of locally constant complex-valued functions on $G_{lcd}$ with compact support.

Such $\mathscr{H}(G)$ algebras play an important role in the theory of decomposition of group representations into tensor products.

Title	Hecke algebra
Canonical name	HeckeAlgebra
Date of creation	2013-03-22 14:08:13
Last modified on	2013-03-22 14:08:13
Owner	olivierfouquetx (2421)
Last modified by	olivierfouquetx (2421)
Numerical id	13
Author	olivierfouquetx (2421)
Entry type	Definition
Classification	msc 11F11
Classification	msc 20C08
Related topic	ModularForms
Related topic	AlgebraicNumberTheory
Defines	Hecke operator
Defines	Hecke algebra $H(G)$ of the group G