The splitting of reductions of an abelian variety

2011/11/02 by David Zywina, Zywina, David · 1 voice
Mathematics · #14K15 (Primary) 11F80 (Secondary) #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F80 #msc:14K15

paper · pdf · doi:10.48550/arxiv.1111.0624

arxiv created 2011/11/02 · arxiv updated 2011/11/03

Abstract

Consider an absolutely simple abelian variety A defined over a number field K. For most places v of K, we study how the reduction A_v of A modulo v splits up to isogeny. Assuming the Mumford-Tate conjecture for A and possibly increasing K, we will show that A_v is isogenous to the m-th power of an absolutely simple abelian variety for all places v of K away from a set of density 0, where m is an integer depending only on the endomorphism ring End(A_Kbar). This proves many cases, and supplies justification, for a conjecture of Murty and Patankar. Under the same assumptions, we will also describe the Galois extension of Q generated by the Weil numbers of A_v for most v.

Discussions

• Let $K$ be a number field, let $A/K$ be an abelian surface, and let $d>0$ be squarefree. Suppose that $\mathbb{Q}(\sqrt{-d}) \not\subset \operatorname{End}^0_{K}(A)$ . Is it possible that $\mathbb{Q}( [stackexchange, 7 points]

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