  Jean-Pierre Serre has conjectured, in the context of abelian varieties, that
there are infinitely primes of good ordinary reduction for a smooth, projective
variety over a number field. We consider this conjecture and its natural
variants. In particular we have conjectured (with C. S. Rajan) the existence of
infinitely many primes of Hodge-Witt reduction (any prime of ordinary reduction
is also a prime of Hodge-Witt reduction). The two conjectures are not
equivalent but are related. We prove a precise relationship between the two; we
prove several results which provide some evidence for these conjectures; we
show that primes of ordinary and Hodge-Witt reduction can have different
densities. We prove our conjecture on Hodge- Witt and ordinary reduction for
abelian varieties with complex multiplication. We include here an unpublished
joint result with C. S. Rajan (also independently established by Fedor
Bogomolov and Yuri Zarhin by a different method) on the existence of primes of
ordinary reductions for K3 surfaces; our proof also shows that for an abelian
threefold over a number field there is a set of primes of positive density at
which it has Hodge-Witt reduction (this is also a joint result with C. S.
Rajan). We give a number of examples including those of Fermat hypersurfaces
for which all the conjectures we make hold.
