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    $\begingroup$ Nice answer! As for the bunny's specific question, you could perhaps add that in the last few years a purely p-adic proof of the Weil conjectures has been completed using rigid cohomology (by Kiran Kedlaya, building on work of others). $\endgroup$ Commented Mar 22, 2010 at 0:30
  • $\begingroup$ "Dwork's difficult masterpiece p-adic cycles" I wonder since some time if I should read that? A Zentralblatt review indicates that it may be outdated, an article by Katz on parts of it is described as unreadable. At this occasion an other question: Do you have a text of your talk: math.uchicago.edu/~emerton/AlgGeom/AlgGeomAbstracts/p-adic.html ? $\endgroup$ Commented Mar 22, 2010 at 11:05