Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics included modular forms, varieties over finite fields, rational and integral points on varieties, class groups, and integer factorization

Compter (rapidement) le nombre de solutions d'\'equations dans les corps finis

The number of solutions in finite fields of a system of polynomial equations obeys a very strong regularity, reflected for example by the rationality of the zeta function of an algebraic variety defined over a finite field, or the modularity of Hasse-Weil's $L$-function of an elliptic curve over \Q. Since two decades, efficient methods have been invented to compute effectively this number of solutions, notably in view of cryptographic applications. This expos\'e presents some of these methods, generally relying on the use of Lefshetz's trace formula in an adequate cohomology theory and discusses their respective advantages. ----- Le nombre de solutions dans les corps finis d'un syst\`eme d'\'equations polynomiales ob\'eit \`a une tr\`es forte r\'egularit\'e, refl\'et\'ee par exemple par la rationalit\'e de la fonction z\^eta d'une vari\'et\'e alg\'ebrique sur un corps fini, ou la modularit\'e de la fonction $L$ de Hasse-Weil d'une courbe elliptique sur \Q. Depuis une vingtaine d'ann\'ees des m\'ethodes efficaces ont \'et\'e invent\'ees pour calculer effectivement ce nombre de solutions, notamment en vue d'applications \`a la cryptographie. L'expos\'e en pr\'esentera quelques-unes, g\'en\'eralement fond\'ees l'utilisation de la formule des traces de Lefschetz dans une th\'eorie cohomologique convenable, et expliquera leurs avantages respectifs.Comment: S\'eminaire Bourbaki, 50e ann\'ee, expos\'e 968, Novembre 2006. 48 pages, in french. Final version to appear in Ast\'erisqu

Computing isogenies between Abelian Varieties

47 pagesInternational audienceWe describe an efficient algorithm for the computation of isogenies between abelian varieties represented in the coordinate system provided by algebraic theta functions. We explain how to compute all the isogenies from an abelian variety whose kernel is isomorphic to a given abstract group. We also describe an analog of Vélu's formulas to compute an isogenis with prescribed kernels. All our algorithms rely in an essential manner on a generalization of the Riemann formulas. In order to improve the efficiency of our algorithms, we introduce a point compression algorithm that represents a point of level $4\ell$ of a $g$ dimensional abelian variety using only $g(g+1)/2\cdot 4^g$ coordinates. We also give formulas to compute the Weil and commutator pairing given input points in theta coordinates. All the algorithms presented in this paper work in general for any abelian variety defined over a field of odd characteristic