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
yj G SL2(Z). Taking
j ) gives in particular
h(z + l) = h(z).
Therefore h defines a holomorphic function of q =
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
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
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
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