Modular Forms: A Classical Approach
Modular Forms: A Classical Approach
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DOI：
10.1090/gsm/179
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发表时间：
2017-08
期刊：
影响因子：
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通讯作者：
H. Cohen;Fredrik Strömberg
中科院分区：
文献类型：
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作者：
H. Cohen;Fredrik Strömberg
模形式理论是数学和物理学许多领域中使用的基本工具。它本身也是一个非常具体和“有趣”的主题，并且充满了惊人数量的令人惊讶的身份。这本综合性的教科书，其中包括许多练习，旨在提供一个完整的图片的经典方面的问题，强调明确的公式。经过一些激励的例子，如椭圆函数和theta函数，模块组，它的子群，以及一般方面的全纯和非全纯模块形式的解释，强调明确的例子。心脏的书是经典理论发展的赫克和继续到阿特金-莱纳-李理论的新形式，包括理论的爱森斯坦系列，兰金-塞尔伯格理论，以及更一般的理论θ系列包括韦尔表示。最后一章详细探讨了更一般的模形式，如半整权、希尔伯特、雅可比、马斯和西格尔模形式。书中的一些“宝石”是Hecke算子的可立即实现的迹公式，Haberland计算Petersson内积的公式的推广，W。李的鲜为人知的定理对角化的全空间的模形式，并明确算法，由于第二作者计算马斯形式。这本书基本上是自成一体的，必要的工具，如伽马和贝塞尔函数，伯努利数，等等被赋予在一个单独的章节。
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 being given in a separate chapter.