Download e-book for iPad: 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

Modular varieties are vastly vital in a variety of parts of arithmetic, from quantity concept and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a customary instance, have deep mathematics value. ahead of this publication, the quickest recognized algorithms for computing those Fourier coefficients took exponential time, other than in a few detailed situations. The case of elliptic curves (Schoof's set of rules) used to be on the beginning of elliptic curve cryptography round 1985. This publication provides an set of rules for computing coefficients of modular varieties of point one in polynomial time. for instance, Ramanujan's tau of a major quantity p should be computed in time bounded via a set strength of the logarithm of p. Such quickly computation of Fourier coefficients is itself in keeping with the most results of the booklet: the computation, in polynomial time, of Galois representations over finite fields connected to modular varieties by means of the Langlands application. simply because those Galois representations generally have a nonsolvable photo, this result's an immense leap forward from specific category box concept, and it may be defined because the commence of the categorical Langlands program.

The computation of the Galois representations makes use of their consciousness, following Shimura and Deligne, within the torsion subgroup of Jacobian different types of modular curves. the most problem is then to accomplish the required computations in time polynomial within the measurement of those hugely nonlinear algebraic kinds. certain computations related to platforms of polynomial equations in lots of variables take exponential time. this can be kept away from via numerical approximations with a precision that suffices to derive specified 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 bought from Arakelov concept. varieties of approximations are taken care of: one utilizing advanced uniformization and one other one utilizing geometry over finite fields.

The ebook starts with a concise and urban creation that makes its available to readers with no an in depth historical past in mathematics geometry. And the e-book encompasses a bankruptcy that describes real computations.

Show description

Read or Download 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) PDF

Similar number theory books

Get Stream Ciphers and Number Theory (North-Holland Mathematical PDF

This is often the original booklet on cross-fertilisations among circulate ciphers and quantity idea. It systematically and comprehensively covers recognized connections among the 2 components which are to be had in basic terms in study papers. a few elements of this ebook include new examine effects that aren't on hand in other places.

Download e-book for kindle: Arithmetic of Quadratic Forms (Springer Monographs in by Goro Shimura

This e-book is split into components. the 1st half is initial and comprises algebraic quantity thought and the idea of semisimple algebras. There are important themes: category of quadratic varieties and quadratic Diophantine equations. the second one subject is a brand new framework which incorporates the research of Gauss at the sums of 3 squares as a unique case.

Theory of Algebraic Integers (Cambridge Mathematical by Richard Dedekind,John Stillwell PDF

The discovery of beliefs through Dedekind within the 1870s used to be good sooner than its time, and proved to be the genesis of what at the present time we'd name algebraic quantity idea. His memoir 'Sur los angeles Theorie des Nombres Entiers Algebriques' first seemed in instalments within the 'Bulletin des sciences mathematiques' in 1877.

An Introduction to the Theory of the Riemann Zeta-Function - download pdf or read online

It is a smooth advent to the analytic recommendations utilized in the research of zeta features, throughout the instance of the Riemann zeta functionality. Riemann brought this functionality in reference to his learn of leading numbers and from this has constructed the topic of analytic quantity idea. due to the fact that then many different periods of 'zeta functionality' were brought and they're now essentially the most intensively studied gadgets in quantity conception.

Additional info for 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)

Sample text

Download PDF sample

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


by Anthony
4.1

Rated 4.16 of 5 – based on 38 votes