Let X be a proper smooth curve over a perfect field of characteristic p > 0 and U an open dense subscheme of X. We prove that convergent F-isocrystals on U are overconvergent under the condition that they are overconvergent at each point in X \ U. Using this criterion, we show that the higher direct images R i f crys * O V by a proper smooth morphism of schemes f : V → U are overconvergent.
Abstract. In this paper, we study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the noncommutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We propose also a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field. This is a preprint which is distributed since 2005 which is still in the process of submision. Following a recent modification of some technical mistakes in the previous version of the paper as well as an amelioration of the presentation of the paper, we decide wider distribution via the archive.
Abstract. We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over Z d p -extensions of function fields ramifying at a finite set of places.
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