The Fourier coefficients of modular forms are
of widespread interest as an important source of arithmetic
information. In many cases, these coefficients can be recovered from
explicit knowledge of the traces of Hecke operators. The original
trace formula for Hecke operators was given by Selberg in 1956. Many
improvements were made in subsequent years, notably by Eichler and
Hijikata.

This book provides a comprehensive modern treatment of the
Eichlerâ€“Selberg/Hijikata trace formula for the traces of Hecke
operators on spaces of holomorphic cusp forms of weight
$\mathtt{k}>2$ for congruence subgroups of
$\operatorname{SL}_2(\mathbf{Z})$. The first half of the
text brings together the background from number theory and
representation theory required for the computation. This includes
detailed discussions of modular forms, Hecke operators, adeles and
ideles, structure theory for $\operatorname{GL}_2(\mathbf{A})$, strong
approximation, integration on locally compact groups, the Poisson
summation formula, adelic zeta functions, basic representation theory
for locally compact groups, the unitary representations of
$\operatorname{GL}_2(\mathbf{R})$, and the connection between
classical cusp forms and their adelic counterparts on
$\operatorname{GL}_2(\mathbf{A})$.

The second half begins with a full development of the geometric
side of the Arthurâ€“Selberg trace formula for the group
$\operatorname{GL}_2(\mathbf{A})$. This leads to an
expression for the trace of a Hecke operator, which is then computed
explicitly. The exposition is virtually self-contained, with complete
references for the occasional use of auxiliary results. The book
concludes with several applications of the final formula.

Readership

Graduate students and research mathematicians interested in
number theory, particularly modular forms, Hecke operators, and trace
formulas.