Computational aspects of modular forms and Galois representations

Details Computational aspects of modular forms and Galois representations Computational aspects of modular forms and Galois representations

2006-05-09

arxiv.org/abs/math/0605244v3

Mathematics

Number Theory

438 pages. This book is to appear in the series 'Annals of Mathematics Studies' of Princeton University Press, after reviewing

Scientific paper

This is a book about computational aspects of modular forms and the Galois representations attached to them. The main result is the following: Galois representations over finite fields attached to modular forms of level one can, in almost all cases, be computed in polynomial time in the weight and the size of the finite field. As a consequence, coefficients of modular forms can be computed fast via congruences, as in Schoof's algorithm for the number of points of elliptic curves over finite fields. The most important feature of the proof of the main result is that exact computations involving systems of polynomial equations in many variables are avoided by approximations and height bounds, i.e., bounds for the accuracy that is necessary to derive exact values from the approximations.

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