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Books > Science & Mathematics > Mathematics > Algebra
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Modular Forms - A Classical Approach (Hardcover)
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Modular Forms - A Classical Approach (Hardcover)
Series: Graduate Studies in Mathematics
Expected to ship within 12 - 19 working days
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The theory of modular forms is a fundamental tool used in many
areas of mathematics and physics. It is also a very concrete and
"fun" subject in itself and abounds with an amazing number of
surprising identities. This comprehensive textbook, which includes
numerous exercises, aims to give a complete picture of the
classical aspects of the subject, with an emphasis on explicit
formulas. After a number of motivating examples such as elliptic
functions and theta functions, the modular group, its subgroups,
and general aspects of holomorphic and nonholomorphic modular forms
are explained, with an emphasis on explicit examples. The heart of
the book is the classical theory developed by Hecke and continued
up to the Atkin-Lehner-Li theory of newforms and including the
theory of Eisenstein series, Rankin-Selberg theory, and a more
general theory of theta series including the Weil representation.
The final chapter explores in some detail more general types of
modular forms such as half-integral weight, Hilbert, Jacobi, Maass,
and Siegel modular forms. Some "gems" of the book are an
immediately implementable trace formula for Hecke operators,
generalizations of Haberland's formulas for the computation of
Petersson inner products, W. Li's little-known theorem on the
diagonalization of the full space of modular forms, and explicit
algorithms due to the second author for computing Maass forms. This
book is essentially self-contained; the necessary tools such as
gamma and Bessel functions, Bernoulli numbers, and so on are given
in a separate chapter.
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