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Computational Aspects of Modular Forms and Galois Representations
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Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Bas Edixhoven and Jean-Marc Couveignes

Abstract

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fix ... More

Keywords: modular forms, arithmetic geometry, computing algorithms, Fourier coefficients, computing coefficients, polynomial time, Ramanujan's tau, Galois representations, Langlands program, computation

Bibliographic Information

Print publication date: 2011 Print ISBN-13: 9780691142012
Published to University Press Scholarship Online: October 2017 DOI:10.23943/princeton/9780691142012.001.0001

Authors

Affiliations are at time of print publication.

Bas Edixhoven, editor
Universiteit Leiden

Jean-Marc Couveignes, editor
Universite de Bordeaux

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