Traces of Hecke Operators
Title | Traces of Hecke Operators PDF eBook |
Author | Andrew Knightly |
Publisher | American Mathematical Soc. |
Total Pages | 392 |
Release | 2006 |
Genre | Mathematics |
ISBN | 0821837397 |
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 Eichler-Selberg/Hijikata trace formulafor 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. Thisincludes 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 $\operatorname{GL 2(\mathbf{A )$. Thesecond half begins with a full development of the geometric side of the Arthur-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, withcomplete references for the occasional use of auxiliary results. The book concludes with several applications of the final formula.
Modular Forms and Hecke Operators
Title | Modular Forms and Hecke Operators PDF eBook |
Author | A. N. Andrianov |
Publisher | American Mathematical Soc. |
Total Pages | 346 |
Release | 2016-01-29 |
Genre | |
ISBN | 1470418681 |
he concept of Hecke operators was so simple and natural that, soon after Hecke's work, scholars made the attempt to develop a Hecke theory for modular forms, such as Siegel modular forms. As this theory developed, the Hecke operators on spaces of modular forms in several variables were found to have arithmetic meaning. Specifically, the theory provided a framework for discovering certain multiplicative properties of the number of integer representations of quadratic forms by quadratic forms. Now that the theory has matured, the time is right for this detailed and systematic exposition of its fundamental methods and results. Features: The book starts with the basics and ends with the latest results, explaining the current status of the theory of Hecke operators on spaces of holomorphic modular forms of integer and half-integer weight congruence-subgroups of integral symplectic groups.Hecke operators are considered principally as an instrument for studying the multiplicative properties of the Fourier coefficients of modular forms. It is the authors' intent that Modular Forms and Hecke Operators help attract young researchers to this beautiful and mysterious realm of number theory.
Traces of Hecke Operators
Title | Traces of Hecke Operators PDF eBook |
Author | Seth Padowitz |
Publisher | |
Total Pages | 172 |
Release | 1998 |
Genre | Automorphic functions |
ISBN |
Modular Forms
Title | Modular Forms PDF eBook |
Author | Toshitsune Miyake |
Publisher | Springer Science & Business Media |
Total Pages | 343 |
Release | 2006-02-17 |
Genre | Mathematics |
ISBN | 3540295933 |
This book is a translation of the earlier book written by Koji Doi and the author, who revised it substantially for this English edition. It offers the basic knowledge of elliptic modular forms necessary to understand recent developments in number theory. It also treats the unit groups of quaternion algebras, rarely dealt with in books; and in the last chapter, Eisenstein series with parameter are discussed following the recent work of Shimura.
Modular Functions of One Variable, I-IV
Title | Modular Functions of One Variable, I-IV PDF eBook |
Author | Willem Kuyk |
Publisher | |
Total Pages | |
Release | 1973 |
Genre | Modular functions |
ISBN |
Hecke Operators on R (N) and Their Traces
Title | Hecke Operators on R (N) and Their Traces PDF eBook |
Author | Peter George Kluit |
Publisher | |
Total Pages | 0 |
Release | 1979 |
Genre | |
ISBN |
The trace formula for Hecke operators over rank one lattices
Title | The trace formula for Hecke operators over rank one lattices PDF eBook |
Author | Werner Hoffmann |
Publisher | |
Total Pages | 113 |
Release | 1986 |
Genre | |
ISBN |