Fonction $L$ de Hasse-Weil d'une Variété Abélienne sur un Corps de Fonction Algébrique à Réduction Semi-stable
Vol. 9 (2002), No. 2, Page 279--301.
Trihan, Fabien
Fonction $L$ de Hasse-Weil d'une Variété Abélienne sur un Corps de Fonction Algébrique à Réduction Semi-stable
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Abstract:
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.
Mathematics Subject Classification (1991): 14F30
Mathematical Reviews Number: MR1904933
Received: 2000-09-21
