Explicit bounds for split reductions of simple abelian varieties

Jeffrey D. Achter[1]

  • [1] Department of Mathematics Colorado State University Fort Collins, CO 80523

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 1, page 41-55
  • ISSN: 1246-7405

Abstract

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Let X / K be an absolutely simple abelian variety over a number field; we study whether the reductions X 𝔭 tend to be simple, too. We show that if End ( X ) is a definite quaternion algebra, then the reduction X 𝔭 is geometrically isogenous to the self-product of an absolutely simple abelian variety for 𝔭 in a set of positive density, while if X is of Mumford type, then X 𝔭 is simple for almost all 𝔭 . For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound for the growth of the set of primes of non-simple reduction.

How to cite

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Achter, Jeffrey D.. "Explicit bounds for split reductions of simple abelian varieties." Journal de Théorie des Nombres de Bordeaux 24.1 (2012): 41-55. <http://eudml.org/doc/251138>.

@article{Achter2012,
abstract = {Let $X/K$ be an absolutely simple abelian variety over a number field; we study whether the reductions $X_\{\mathfrak\{p\}\}$ tend to be simple, too. We show that if $\operatorname\{End\}(X)$ is a definite quaternion algebra, then the reduction $X_\{\mathfrak\{p\}\}$ is geometrically isogenous to the self-product of an absolutely simple abelian variety for $\{\mathfrak\{p\}\}$ in a set of positive density, while if $X$ is of Mumford type, then $X_\{\mathfrak\{p\}\}$ is simple for almost all $\{\mathfrak\{p\}\}$. For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound for the growth of the set of primes of non-simple reduction.},
affiliation = {Department of Mathematics Colorado State University Fort Collins, CO 80523},
author = {Achter, Jeffrey D.},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {3},
number = {1},
pages = {41-55},
publisher = {Société Arithmétique de Bordeaux},
title = {Explicit bounds for split reductions of simple abelian varieties},
url = {http://eudml.org/doc/251138},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Achter, Jeffrey D.
TI - Explicit bounds for split reductions of simple abelian varieties
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/3//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 1
SP - 41
EP - 55
AB - Let $X/K$ be an absolutely simple abelian variety over a number field; we study whether the reductions $X_{\mathfrak{p}}$ tend to be simple, too. We show that if $\operatorname{End}(X)$ is a definite quaternion algebra, then the reduction $X_{\mathfrak{p}}$ is geometrically isogenous to the self-product of an absolutely simple abelian variety for ${\mathfrak{p}}$ in a set of positive density, while if $X$ is of Mumford type, then $X_{\mathfrak{p}}$ is simple for almost all ${\mathfrak{p}}$. For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound for the growth of the set of primes of non-simple reduction.
LA - eng
UR - http://eudml.org/doc/251138
ER -

References

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