## Fonction $L$ de Hasse-Weil d'une VariÃ©tÃ© AbÃ©lienne sur un Corps de Fonction AlgÃ©brique Ã  RÃ©duction Semi-stable

J. Math. Sci. Univ. Tokyo
Vol. 9 (2002), No. 2, Page 279--301.

Trihan, Fabien
Fonction $L$ de Hasse-Weil d'une VariÃ©tÃ© AbÃ©lienne sur un Corps de Fonction AlgÃ©brique Ã  RÃ©duction Semi-stable
In the Birch/Swinnerton-Dyer conjecture, one studies the behaviour of the Hasse-Weil $L$-function of an abelian variety. In the case of function fields of characteristic $p>0$, one needs to know the $p$-adic valuation of the leading coefficient of this function at $s=1$ and therefore, needs a Grothendieck-type cohomological interpretation of it in terms of a "good" $p$-adic cohomology (integral if possible) as it was shown in [Ba] in the good reduction case. We give such interpretation in the semi-abelian, semi-stable and then potentially semi-abelian case and obtain as a bonus a description of the cohomology of a quasi-unipotent isocrystal (in the sense of [Cr4]) over a smooth affine curve.