American Institute of Mathematical Sciences

November  2014, 8(4): 511-519. doi: 10.3934/amc.2014.8.511

Splitting of abelian varieties

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

Received  February 2014 Revised  September 2014 Published  November 2014

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 $V$.
Citation: V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511
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References:
 [1] Fang Li, Jie Pan. On inner Poisson structures of a quantum cluster algebra without coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021021

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