Download e-book for kindle: Computational Aspects of Modular Forms and Galois by Bas Edixhoven,Jean-Marc Couveignes,Robin de Jong,Franz

By Bas Edixhoven,Jean-Marc Couveignes,Robin de Jong,Franz Merkl,Johan Bosman

ISBN-10: 0691142017

ISBN-13: 9780691142012

ISBN-10: 0691142025

ISBN-13: 9780691142029

Modular kinds are drastically vital in numerous components of arithmetic, from quantity idea and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a regular instance, have deep mathematics importance. ahead of this publication, the quickest identified algorithms for computing those Fourier coefficients took exponential time, other than in a few unique situations. The case of elliptic curves (Schoof's set of rules) used to be on the start of elliptic curve cryptography round 1985. This publication supplies an set of rules for computing coefficients of modular kinds of point one in polynomial time. for instance, Ramanujan's tau of a first-rate quantity p may be computed in time bounded via a set energy of the logarithm of p. Such quick computation of Fourier coefficients is itself in response to the most results of the ebook: the computation, in polynomial time, of Galois representations over finite fields hooked up to modular varieties by way of the Langlands software. simply because those Galois representations generally have a nonsolvable picture, this result's an incredible breakthrough from specific type box concept, and it may be defined because the begin of the specific Langlands program.

The computation of the Galois representations makes use of their attention, following Shimura and Deligne, within the torsion subgroup of Jacobian types of modular curves. the most problem is then to accomplish the mandatory computations in time polynomial within the measurement of those hugely nonlinear algebraic kinds. certain computations regarding platforms of polynomial equations in lots of variables take exponential time. this can be kept away from by way of numerical approximations with a precision that suffices to derive distinctive effects from them. Bounds for the necessary precision--in different phrases, bounds for the peak of the rational numbers that describe the Galois illustration to be computed--are acquired from Arakelov concept. varieties of approximations are taken care of: one utilizing complicated uniformization and one other one utilizing geometry over finite fields.

The booklet starts off with a concise and urban creation that makes its available to readers with out an intensive history in mathematics geometry. And the booklet encompasses a bankruptcy that describes genuine computations.

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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) (Annals of Mathematics Studies) by Bas Edixhoven,Jean-Marc Couveignes,Robin de Jong,Franz Merkl,Johan Bosman


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