The splitting of reductions of an abelian variety (Q2927905)

Let \(A\) be an absolutely simple abelian variety defined over a number field \(K\) such that all \(\ell\)-adic monodromy groups associated to \(A\) over \(K\) are connected. Consider the integer \(m:= [\text{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}: E]^{{1\over 2}}\), where \(E\) is the center of the division algebra \(\text{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}\).NEWLINENEWLINE The main result of the paper is the following theorem: For all places \(v\) of \(K\) away from a set of density \(0\), \(A_v\) is isogenous to \(B^m\) for some abelian variety \(B\) over \(\mathbb{F}_v\). Under the assumption that the Mumford-Tate conjecture holds for \(A\), the abelian variety \(B\) is absolutely simple. This implies the following conjecture due to Murty-Patankar: Let \(A\) be as above and let \({\mathcal V}\) be the set of finite places \(v\) of \(K\) for which \(A\) has good reduction and \(A_v\) over \(\mathbb{F}_v\) is simple. Then, after possibly replacing \(K\) by a finite extension, \({\mathcal V}\) has density \(1\) if and only if \(\text{End}(A_{\overline K})\) is commutative. Under the same assumptions, the paper also describes the Galois extensions of \(\mathbb{Q}\) generated by the Weil numbers of \(A_v\) for most \(v\).