Let $$\kappa $$ κ be a field, finitely generated over its prime field, and let k denote an algebraically closed field containing $$\kappa $$ κ . For a perverse $$\overline{\mathbb {Q}}_\ell $$ Q ¯ ℓ -adic sheaf $$K_0$$ K 0 on an abelian variety $$X_0$$ X 0 over $$\kappa $$ κ , let K and X denote the base field extensions of $$K_0$$ K 0 and $$X_0$$ X 0 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. $$\chi (X,K)=\sum _\nu (-1)^\nu \dim _{\overline{\mathbb {Q}}_\ell }(H^\nu (X,K))\ge 0$$ χ ( X , K ) = ∑ ν ( - 1 ) ν dim Q ¯ ℓ ( H ν ( X , K ) ) ≥ 0 . This generalizes the result of Franecki and Kapranov [9] for fields of characteristic zero. Furthermore we show that $$\chi (X,K)=0$$ χ ( X , K ) = 0 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.