Ordinary reductions of abelian varieties
  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). 
Kirti Joshi
math.AG math.NT