Root numbers of abelian varieties

We generalize a theorem of D. Rohrlich concerning root numbers of elliptic curves over number fields. Our result applies to arbitrary abelian varieties. Namely, under certain conditions which naturally extend the conditions used by D. Rohrlich, we show that the root number $W(A,\tau)$ associated to an abelian variety $A$ over a number field $F$ and a complex finite-dimensional irreducible representation $\tau$ of $\operatorname{Gal}(\overline{F}/F)$ with real-valued character is equal to $1$. We also show that our result is consistent with a refined version of the conjecture of Birch and Swinnerton-Dyer.