  We partially answer, in terms of monodromy, Murty and Patankar's question:
Given an absolutely simple abelian variety over a number field, does it have
simple specializations at a set of places of positive Dirichlet density?
  The answer 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.
