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EXPLICIT BOUNDS FOR SPLIT REDUCTIONS OF SIMPLE ABELIAN JEFFREY D. ACHTER
 

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 Mumford-Tate 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 Mumford-Tate 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

  

Source: Achter, Jeff - Department of Mathematics, Colorado State University

 

Collections: Environmental Sciences and Ecology; Mathematics