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)
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)
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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 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.
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Front Matter
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One
Introduction, main results, context
Bas Edixhoven
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Two
Modular curves, modular forms, lattices, Galois representations
Bas Edixhoven
- Three First description of the algorithms
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Four
Short introduction to heights and Arakelov theory
Bas Edixhoven andRobin de Jong
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Five
Computing complex zeros of polynomials and power series
Jean-Marc Couveignes
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Six
Computations with modular forms and Galois representations
Johan Bosman
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Seven
Polynomials for projective representations of level one forms
Johan Bosman
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Eight
Description of X1(5l)
Bas Edixhoven
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Nine
Applying Arakelov theory
Bas Edixhoven andRobin de Jong
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Ten
An upper bound for Green functions on Riemann surfaces
Franz Merkl
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Eleven
Bounds for Arakelov invariants of modular curves
Bas Edixhoven andRobin de Jong
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Twelve
Approximating Vf over the complex numbers
Jean-Marc Couveignes
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Thirteen
Computing Vf modulo p
Jean-Marc Couveignes
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Fourteen
Computing the residual Galois representations
Bas Edixhoven
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Fifteen
Computing coefficients of modular forms
Bas Edixhoven
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Epilogue
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End Matter
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