Splitting of abelian varieties

V. Kumar Murty; Ying Zong

doi:10.3934/amc.2014.8.511

Article Contents

2014, Volume 8, Issue 4: 511-519. Doi: 10.3934/amc.2014.8.511

This issue Previous Article On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$ Next Article

Splitting of abelian varieties

V. Kumar Murty^1, and
Ying Zong^1,

1.
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Canada M5S 2E4

Received: January 31, 2014

Revised: August 31, 2014

Published: October 2014

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Abstract

It is possible that a simple (or absolutely simple) Abelian variety defined over a number field splits modulo every prime of good reduction. This is a new problem that arises in designing crypto systems using Abelian varieties of dimension larger than $1$. We discuss what is behind this phenomenon. In particular, we discuss the question of given an absolutely simple abelian variety over a number field, whether it has simple specializations at a set of places of positive Dirichlet density? A conjectural answer to this question was given by Murty and Patankar, and we explain some recent progress towards proving the conjecture. Our result ([14], Theorem 1.1) is based on the classification of pairs $(G,V)$ consisting of a semi-simple algebraic group $G$ over a non-archimedean local field and an absolutely irreducible representation $V$ of $G$ such that $G$ admits a maximal torus acting irreducibly on $V$.

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Mathematics Subject Classification: 11G10, 11G25, 14K15.

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