Split reductions of simple abelian varieties (Q843036): Difference between revisions

\textit{V. Kumar Murty} and \textit{V. M. Patankar} [Int. Math. Res. Not. 2008, Article ID rnn033, 27 p. (2008; Zbl 1152.14043)] study the splitting behavior of abelian varieties over number field, and advance the following Conjecture. Let \(X/K\) be an absolutely simple abelian variety over a number field. The set of primes of \(K\) where \(X\) split has positive density if and only if \(\mathrm{End}_{\overline{K}}(X)\) is noncommutative. Theorem A. Let \(X/K\) be an absolutely simple abelian variety over a number field. Suppose that either (i) \(\mathrm{End}_{\overline{K}}(X) \otimes {\mathbb{Q}} \cong F\) a totally real field, and \(\dim X/[F:{\mathbb{Q}}]\) is odd; or (ii) \(\mathrm{End}_{\overline{K}}(X) \otimes {\mathbb{Q}} \cong E\) a totally imaginary field, and the action of \(E\) on \(X\) is not special. Then for almost every prime \(p,\) \(X_{p}\) is absolutely simple. Theorem B. Suppose \(X/K\) is an absolutely simple abelian variety over a number field, and that \(\mathrm{End}_{\overline{K}}(X)\) is noncommutative. (i) For \(p\) in a set of positive density, \(X_{p}\) absolutely reducible. (ii) Suppose \(\mathrm{End}_{\overline{K}}(X) \otimes {\mathbb{Q}}\) is an indefinite quaternion algebra over a totally real field \(F,\) and that \(\dim X/2[F:{\mathbb{Q}}]\) is odd. For \(p\) in a set of positive density, \(X_{p}\) is geometrically isogenous to the self-product of an absolutely simple abelian variety.