Éditeur(s) : HAL CCSD
Résumé : In this paper we define the $p$-density of a finite subset $Dsubsetma{N}^r$, and show that it gives a good lower bound for the $p$-adic valuation of exponential sums over finite fields of characteristic $p$. We also give an application: when $r=1$, the $p$-density is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in $D$.
https://hal.archives-ouvertes.fr/hal-00551478

hal-00551478
https://hal.archives-ouvertes.fr/hal-00551478
ARXIV : 0812.3382

Éditeur(s) : HAL CCSD
Résumé : In this paper we compute the first vertex of a generic Newton polygon in some special cases, and the corresponding Hasse polynomial. This allows us to show the nonexistence of $p$-cyclic coverings of the projective line in characteristic $p$ with supersingular jacobian for some (infinite families of) genera.
https://hal.archives-ouvertes.fr/hal-00551456

hal-00551456
https://hal.archives-ouvertes.fr/hal-00551456
ARXIV : 0912.2051

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We show that, for any finite field Fq , there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.
ISSN: 0022-314X

hal-00978911
https://hal.archives-ouvertes.fr/hal-00978911
https://hal.archives-ouvertes.fr/hal-00978911/document
https://hal.archives-ouvertes.fr/hal-00978911/file/AubryBlache5_10.pdf
DOI : 10.1016/j.jnt.2007.10.014

Éditeur(s) : HAL CCSD
Résumé : accepté pour publication par l'International Journal of Number Theory
In this paper, we precise the asymptotic behaviour of Newton polygons of $L$-functions associated to character sums, coming from certain $n$ variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum $Delta=Delta_1oplus Delta_2$ when we know the limit of generic Newton polygons for each factor. To our knowledge, these are the first results concerning the asymptotic behaviour of Newton polygons for multivariable polynomials when the generic Newton polygon differs from the combinatorial (Hodge) polygon associated to the polyhedron.
https://hal.archives-ouvertes.fr/hal-00551461

Droits : info:eu-repo/semantics/OpenAccess
hal-00551461
https://hal.archives-ouvertes.fr/hal-00551461
https://hal.archives-ouvertes.fr/hal-00551461/document
https://hal.archives-ouvertes.fr/hal-00551461/file/prodpolytopespubli.pdf

Éditeur(s) : HAL CCSD
Résumé : Dans ce mémoire, on se propose de décrire certains résultats de l'auteur sur les propriétés $p$-adiques des fonctions $L$ associées à des caractères sur les corps finis, à la suite des travaux de Dwork, Robba, Adolphson et Sperber, Wan, entre autres. On parlera aussi de sommes de caractères (et de leurs fonctions $L$) définies sur certains anneaux locaux.
https://tel.archives-ouvertes.fr/tel-00440335

Droits : info:eu-repo/semantics/OpenAccess
tel-00440335
https://tel.archives-ouvertes.fr/tel-00440335
https://tel.archives-ouvertes.fr/tel-00440335/document
https://tel.archives-ouvertes.fr/tel-00440335/file/hdrsommescarfinal.pdf

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Let C be a smooth curve over a Galois ring R. Let f be a function over C, and Ψ be an additive character of order p^l over R; in this paper we study the exponential sums associated to f and Ψ over C, and their L-functions. We show the rationality of the L-functions in a more general setting, then in the case of curves we express them as products of L-functions associated to polynomials over the affine line, each factor coming from a singularity of f. Finally we show that in the case of Morse functions (i.e. having only simple singularities), the degree of the L-functions are, up to sign, the same as in the case of finite fields, yielding very similar bounds for exponential sums.
ISSN: 1071-5797

hal-00551470
https://hal.archives-ouvertes.fr/hal-00551470
https://hal.archives-ouvertes.fr/hal-00551470/document
https://hal.archives-ouvertes.fr/hal-00551470/file/Charlefinal.pdf
DOI : 10.1016/j.ffa.2009.01.001