Galois Invariance of Local Root Numbers

It is a standard remark that the group Aut(C) of abstract field automorphisms of C has uncountably many elements but only two that are are continuous. One consequence is that the analytic properties of a Dirichlet series ∑ n>1 a(n)n −s do not usually carry over to ∑ n>1 ι(a(n))n −s for ι ∈ Aut(C). Silly counterexamples abound: for instance, take a(n) = (1 − √ 2) and choose ι so that ι( √ 2) = − √ 2; then ∑ n>1 a(n)n −s converges everywhere but ∑ n>1 ι(a(n))n −s nowhere. On the other hand, suppose that ∑ n>1 a(n)n −s is the Dirichlet series L(s,M) associated to a motive M over a number field (and to our implicit complex embedding of the coefficient field of M). In this setting one does expect the Dirichlet series L(s,M, ι) = ∑ n>1 ι(a(n))n −s to share certain analytic properties with L(s,M) (= L(s,M, id)). A case in point is the conjecture of Deligne and Gross: If M is pure of odd weight then the order of vanishing of L(s,M, ι) at the center of the critical strip is independent of ι ([3], p. 323, Conjecture 2.7, part (ii)). This conjecture has a number of elementary consequences; for example, it enables one to deduce a “Birch and Swinnerton-Dyer conjecture with twists” from the usual version (cf. [10], p. 127). In this note we examine the consequences for root numbers. The connection with root numbers depends on one further hypothesis. We have already assumed that the weight w of M is odd and hence that the center of the critical strip (w+ 1)/2 is an integer, but suppose that in addition, M is essentially self-dual. In other words, suppose that M∨ ∼= M(w), where M∨ and M(w) are respectively the dual and the w-fold Tate twist of M . The conjectured functional equation of L(s,M, ι) is then a relation bewtween L(s,M, ι) and itself, whence the root number W (M, ι) satisfies