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Elementary Dirichlet Series and Modular Forms Book

Elementary Dirichlet Series and Modular Forms
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Elementary Dirichlet Series and Modular Forms, The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books re, Elementary Dirichlet Series and Modular Forms
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  • Elementary Dirichlet Series and Modular Forms
  • Written by author Goro Shimura
  • Published by Springer-Verlag New York, LLC, 11/19/2010
  • The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books re
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Preface     V
Introduction     1
Preliminaries on Modular Forms and Dirichlet Series     5
Basic symbols and the definition of modular forms     5
Elementary Fourier analysis     13
The functional equation of a Dirichlet series     19
Critical Values of Dirichlet L-functions     25
The values of elementary Dirichlet series at integers     25
The class number of a cyclotomic field     39
Some more formulas for L(k, x)     45
The Case of Imaginary Quadratic Fields and Nearly Holomorphic Modular Forms     53
Dirichlet series associated with an imaginary quadratic field     53
Nearly holomorphic modular forms     55
Eisenstein Series     59
Fourier expansion of Eisenstein series     59
Polynomial relations between Eisenstein series     66
Recurrence formulas for the critical values of certain Dirichlet series     75
Critical Values of Dirichlet Series Associated with Imaginary Quadratic Fields     79
The singular values of nearly holomorphic forms     79
The critical values of L-functions of an imaginary quadratic field     84
The zeta function of a member of a one-parameter family of elliptic curves     96
Supplementary Results     113
Isomorphism classes of abelian varieties with complex multiplication     113
The general case     113
The case of elliptic curves     117
Holomorphic differential operators on the upper half plane     120
Appendix     127
Integration and differentiation under the integral sign     127
Fourier series with parameters     130
The confluent hypergeometric function     131
The Weierstrass [Characters not reproducible]-function     136
The action of G[subscript A+] on modular forms     139
References     145
Index     147


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Elementary Dirichlet Series and Modular Forms, The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books re, Elementary Dirichlet Series and Modular Forms

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Elementary Dirichlet Series and Modular Forms, The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books re, Elementary Dirichlet Series and Modular Forms

Elementary Dirichlet Series and Modular Forms

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Elementary Dirichlet Series and Modular Forms, The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books re, Elementary Dirichlet Series and Modular Forms

Elementary Dirichlet Series and Modular Forms

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