Modular Forms

In this essential, Claudia Alfes-Neumann discusses applications of the theory of modular forms and their importance as fundamental tools in mathematics. These functions - initially defined purely analytically - appear in many areas of mathematics: very prominently in number theory, but also in geometry, combinatorics, representation theory, and physics. After explaining necessary basics from complex analysis, the author defines modular forms and shows some applications in number theory. Furthermore, she takes up two important aspects of the theory surrounding modular forms: Hecke operators and L-functions of modular forms. The essentials conclude with an outlook on real-analytic generalizations of modular forms, which play an important role in current research. This Springer essential is a translation of the original German 1st edition essentials, Modulformen by Claudia Alfes-Neumann, published by Springer Fachmedien Wiesbaden GmbH, part of Springer Nature in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.