Approximations simultanées de deux séries formelles quadratiques
In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic number field. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for factoring integers. Applying a variant of a standard algorithm for finding rings of integers, one finds a subring of the number field...
Arakelov invariants of arithmetic surfaces are well known for genus 1 and 2 ([4], [2]). In this note, we study the modular height and the Arakelov self-intersection for a family of curves of genus 3 with many automorphisms:Arakelov calculus involves both analytic and arithmetic computations. We express the periods of the curve in terms of elliptic integrals. The substitutions used in these integrals provide a splitting of the jacobian of as a product of three elliptic curves. Using the corresponding...
Nous tentons, dans ce survol, de présenter une structure méconnue : l'algèbre de Lie ARI et son groupe GARI. Puis nous montrons quels progrès elle a déjà permis de réaliser dans l'étude arithmético-algébrique des valeurs zêta multiples et aussi quelles possibilités elle ouvre pour l'exploration du phénomène plus général de /emph{dimorphie numérique}.