2008/03/11 by Jean-Yves Étesse, Jean-Yves Etesse, Etesse, Jean-Yves · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #math.AG #msc:11G #msc:11S40 #msc:13B22 #msc:13B35 #msc:13B40 #msc:14F20 #msc:14F30 #msc:14G10 #msc:14G22 #msc:14K #msc:14L05 #msc:14L15
paper · pdf · doi:10.48550/arxiv.0803.1580
arxiv created 2008/03/11 · arxiv updated 2009/12/01
Let k be a perfect field of characteristic p>0, V a complete discrete valuation ring with residue field k and field of fractions K of characteristic 0, and S a separated k-scheme of finite type. When S is smooth over k, we partially prove here a conjecture of Berthelot about the overconvergence of the higher direct images of the structure sheaf under a proper smooth morphism f:X→ S; when k is perfect and V is tamely ramified such direct images are always convergent, not only for the structure sheaf but also for (almost) every convergent F-isocrystals. More generally, we prove this overconvergence when f is liftable over V, or when X is a relative complete intersection in some projective spaces over S, and taking as coefficients any overconvergent isocrystals. We then apply these results to L-functions with coefficients such direct images with Frobenius structure: we derive rationality or meromorphy for these L-functions (Dwork's conjecture), and we study their p-adic unit zeroes and poles (Katz's conjecture) ; and explicit case concerns the ordinary abelian schemes. A more precise presentation of results by chapters is given in the introduction.