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
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 fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The book begins with a concise and concrete introduction that makes it accessible to readers without an extensive background in arithmetic geometry, and it includes a chapter that describes actual computations.
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 Princeton 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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