Elliptic and Modular Functions from Gauss to Dedekind to Hecke

A thorough guide to elliptic functions and modular forms that demonstrates the relevance and usefulness of historical sources.

Many mathematicians today believe mathematical works from past centuries are inaccessible, not useful, or both. This book demonstrates that past works are not only readable, but brimming with ideas and insights. It is aimed at graduate students or researchers working in modular functions, number theory, complex analysis, or special functions.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.