  I show that a conjecture of Joshi-Rajan on primes of Hodge-Witt reduction and
in particular a conjecture of Jean-Pierre Serre on primes of good, ordinary
reduction for an abelian variety over a number field follows from a certain
conjecture on Galois rep- resentations which may perhaps be easier to prove
(and I prove this conjecture for abelian compatible systems of a suitable
type). This reduction (to a conjecture about certain sys- tems of Galois
representations) is based on a new slope estimate for non Hodge-Witt abelian
varieties. In particular for any abelian variety over a number field with at
least one prime of good ordinary or split toric reduction, I show that the
conjecture of Joshi-Rajan and the conjecture of Serre on ordinary reductions
can be reduced to proving that a certain rational trace of Frobenius is in fact
an integer. The assertion that this trace is an integer is proved for abelian
systems of Galois representations (of suitable type).
