Root numbers of abelian varieties and *representations of the Weil -Deligne group

We generalize a theorem of D. Rohrlich concerning root numbers of elliptic curves over the field of rational numbers. Our result applies to abelian varieties over number fields. Namely, under certain conditions which naturally extend the conditions used by D. Rohrlich, we show that the root number W(A, τ) associated to an abelian variety A over a number field F and a complex finite-dimensional irreducible representation τ of Gal(F¯/F) with real-valued character is equal to 1. In the case where the ground field is [special characters omitted], we show that our result is consistent with a refined version of the conjecture of Birch and Swinnerton-Dyer. We also give a description of unitary, orthogonal, and symplectic admissible representations of the Well-Deligne group of a local non-Archimedean field