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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 -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.
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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 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 VĂ©lu'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 of a dimensional abelian variety using only 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