 
Summary: EXPLICIT BOUNDS FOR SPLIT REDUCTIONS OF SIMPLE ABELIAN
VARIETIES
JEFFREY D. ACHTER
Let X/K be an absolutely simple abelian variety over a number field. Evidence
indicates that whether or not X has absolutely simple reduction almost everywhere
depends on the endomorphism ring End(X). On one hand, if End(X) is an indefi
nite division algebra, then every good reduction Xp is actually split, i.e., isogenous
to a product of abelian varieties of smaller dimension [21, Thm. 2(e)].()
On the
other hand, if End(X) is trivial and dim(X) is odd [3], or if X has complex multi
plication [12], then Xp is almost always simple.
Inspired by this, Murty and Patankar conjectured [12] that in general, X has
simple reduction almost everywhere if and only if EndK(X) is commutative.
Using known instances of the MumfordTate conjecture and ideas of Chavdarov
[4], the author made progress [1] towards affirming this conjecture. Moreover, a
preprint of [1] elicited two further questions. W. Gajda shared a preprint of his
work with Banaszak and KrasoŽn on the MumfordTate conjecture for abelian vari
eties of type III [2], and inquired whether it might be used to refine the results of [1]
in that case. V.K. Murty asked whether one has any control over the size of the ex
ceptional set of primes where an abelian variety with commutative endomorphism
