Split reductions of simple abelian varieties

@article{Achter2008SplitRO,
  title={Split reductions of simple abelian varieties},
  author={Jeff Achter},
  journal={Mathematical Research Letters},
  year={2008},
  volume={16},
  pages={199-213}
}
  • Jeff Achter
  • Published 27 June 2008
  • Mathematics
  • Mathematical Research Letters
Consider an absolutely simple abelian variety $X$ over a number field $K$. We show that if the absolute endomorphism ring of $X$ is commutative and satisfies certain parity conditions, then $X_\idp$ is absolutely simple for almost all primes $\idp$. Conversely, if the absolute endomorphism ring of $X$ is noncommutative, then $X_\idp$ is reducible for $\idp$ in a set of positive density. 
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References

SHOWING 1-10 OF 48 REFERENCES
On the image of Galois $l$-adic representations for abelian varieties of type III
In this paper we investigate the image of the l-adic representation attached to the Tate module of an abelian variety defined over a number field. We consider simple abelian varieties of type III in
Splitting of abelian varieties: a new local-global problem
Given a simple or absolutely simple Abelian variety over a number field, do there exist infinitely many places where the reduction remains simple or absolutely simple? In general, the answer is no.
Weil Numbers Generated by Other Weil Numbers and Torsion Fields of Abelian Varieties
Using properties of the Frobenius eigenvalues, we show that, in a precise sense, ‘most’ isomorphism classes of (principally polarized) simple abelian varieties over a finite field are characterized,
A note on good reduction of simple Abelian varieties
In this note it is shown that the reduction of a simple abelian variety of dimension > 2, defined over an algebraic number field, at any finite good prime need not be simple. We give an example of a
On the image of l-adic Galois representations for abelian varieties of type I and II
In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert
Non‐simple abelian varieties in a family: geometric and analytic approaches
We consider, in the special case of certain one‐parameter families of Jacobians of curves defined over a number field, the problem of how the property that the generic fiber of such a family is
Hodge Cycles on Abelian Varieties
The main result proved in these notes is that any Hodge cycle on an abelian variety (in characteristic zero) is an absolute Hodge cycle — see §2 for definitions and (2.11) for a precise statement of
The large sieve and Galois representations
We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we
On the l-adic representations attached to simple abelian varieties of type IV
The l-adic representations associated to prime dimensional type IV absolutely simple abelian varieties over number fields are studied. The image of such a representation was computed. The results
...
...