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Let $X/K$ be an absolutely simple abelian variety over a number field; we study whether the reductions $X_{𝔭}$ tend to be simple, too. We show that if $End\left(X\right)$ is a definite quaternion algebra, then the reduction $X_{𝔭}$ is geometrically isogenous to the self-product of an absolutely simple abelian variety for $𝔭$ in a set of positive density, while if $X$ is of Mumford type, then $X_{𝔭}$ is simple for almost all $𝔭$. For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound...

We estimate the proportion of function fields satisfying certain conditions which imply a function field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even jacobians) over a finite field which have a rational point of order $\ell$.