Hodge level 1 over a p-adic field implies congruence for the number of
rational points of the reduction
(joint -in progress-with Pierre Berthelot and Kay R"ulling)
Abstract: If X is regular and projective over R, the ring of integers of a
p-adic field K with residue field k, with X_K smooth with Hodge level 1, then
the number of rational points of X_k is 1 modulo |k|. If we replace the Hodge
level condition by the l-adic coniveau condition, I had shown already a while
ago that one has the same conclusion. The l-adic coniveau condition implies
the Hodge level condition. Vice-versa, it would be a consequence of the
generalized Hodge conjecture. Needless to say, we have nothing to contribute
to this...We could bypass it to show the theorem.