Let be a field, finitely generated over its prime field, and let k denote an algebraically closed field containing . For a perverse -adic sheaf on an abelian variety over , let K and X denote the base field extensions of and to k. Then, the aim of this note is to show that the Euler-Poincare characteristic of the perverse sheaf K on X is a non-negative integer, i.e. . This generalizes the result of Franecki and Kapranov [9] for fields of characteristic zero. Furthermore we show that implies K to be translation invariant. This result allows to considerably simplify the proof of the generic vanishing theorems for constructible sheaves on complex abelian varieties of [11]. Furthermore it extends these vanishing theorems to constructible sheaves on abelian varieties over finite fields.
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