Root numbers of curves

Abstract

We generalize a theorem of D. Rohrlich concerning root numbers of elliptic curves over the field of rational numbers. Our result applies to curves of all higher genera over number fields. Namely, under certain conditions which naturally extend the conditions used by D. Rohrlich, we show that the root number associated to a smooth projective curve over a number field F and a complex finite-dimensional irreducible representation of the absolute Galois group of F with real-valued character is equal to 1. In the case where the ground field is the field of rational numbers, we show that our result is consistent with the refined version of the conjecture of Birch and Swinnerton-Dyer.