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.