Elliptic and Modular Functions from Gauss to Dedekind to HeckeCambridge University Press, 18 apr 2017 This thorough work presents the fundamental results of modular function theory as developed during the nineteenth and early-twentieth centuries. It features beautiful formulas and derives them using skillful and ingenious manipulations, especially classical methods often overlooked today. Starting with the work of Gauss, Abel, and Jacobi, the book then discusses the attempt by Dedekind to construct a theory of modular functions independent of elliptic functions. The latter part of the book explains how Hurwitz completed this task and includes one of Hurwitz's landmark papers, translated by the author, and delves into the work of Ramanujan, Mordell, and Hecke. For graduate students and experts in modular forms, this book demonstrates the relevance of these original sources and thereby provides the reader with new insights into contemporary work in this area. |
Inhoudsopgave
1 | |
5 | |
13 | |
Abel and Jacobi on Elliptic Functions | 42 |
5 | 132 |
6 | 149 |
7 | 188 |
8 | 212 |
The Theory of Modular Forms as Reworked by Hurwitz | 334 |
Ramanujans Euler Products and Modular Forms | 344 |
Dirichlet Series and Modular Forms | 371 |
Sums of Squares | 384 |
The Hecke Operators | 426 |
Translation of Hurwitzs Paper of 1904 | 445 |
463 | |
471 | |
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Elliptic and Modular Functions from Gauss to Dedekind to Hecke Ranjan Roy Gedeeltelijke weergave - 2017 |
Veelvoorkomende woorden en zinsdelen
Abel algebraic applied arithmetical Cauchy Cauchy’s coefficients complex conjectured constant convergent cusp form Dedekind sums defined denoted derive differential equation Dirichlet series divisors Eisenstein series elliptic functions elliptic integral equation of order equivalent Euler product expressed finite form of weight formula Fourier fundamental domain Gauss gave given Glaisher Hardy Hecke Hecke’s Hermite Hurwitz hypergeometric identity implies infinite product invariant Jacobi Klein Kronecker linear mathematician mathematics matrix method modular equation modular forms modular functions modular group Mordell multiplier equation Note number of representations number theory Observe obtain paper periods polynomial positive integer proof proved quadratic Ramanujan result Riemann right-hand side root of unity satisfied showed solutions sums of squares theorem theory of elliptic theta functions transformation upper half-plane values Weierstrass wrote zero