Traces of Hecke operators

1. Introduction

A modular form of level 1 and weight k is a holomorphic function h(z)

on the complex upper half-plane H which satisfies

M ^ ) = (cz + dfh{z)

cz + a

for all

(a

yj G SL2(Z). Taking

(a

j ) — gives in particular

h(z + l) = h(z).

Therefore h defines a holomorphic function of q =

e27rlz.

The mapping

z i— q takes H onto the open unit disk with the origin removed. The

origin corresponds to the cusp z — zoo. Modular forms are required to

be holomorphic at the cusps, i.e. as a function of q, h has a power series

expansion

oo

If ao = 0, then h is a cusp form. The Fourier coefficients an of modu-

lar forms contain a great deal of arithmetic information. For instance the

following quantities are encoded in the Fourier coefficients of appropriately

chosen modular forms:

• The number of ways of representing an integer by a given quadratic

form, e.g. as a sum of four squares ([Iwl], Ch. 10, 11.)

• The number of points on a Q-rational elliptic curve over the field

with p elements. (See the survey [Da].)

One way to access the Fourier coefficients is as follows. For each prime

number p (not dividing the level N) there is a linear Hecke operator Tp

acting on the vector space of cusp forms of a given level and weight. Tp

is diagonalizable, and its eigenvalues are the

pth

Fourier coefficients of the

elements of a certain basis of eigenvectors. There is a well-known formula

for the trace of Tp from which these Fourier coefficients can be recovered.

This formula was originally given in the level 1 case by Selberg without

proof in his pioneering 1956 paper [S] on the trace formula for SL2(R).

Subsequent improvements were made by various authors, notably Eichler

[E], who developed a different technique allowing k = 2 and square-free

I

http://dx.doi.org/10.1090/surv/133/01 http://dx.doi.org/10.1090/surv/133/01