Galois Theory, Elliptic Curves, and Root Numbers

The inverse problem of Galois theory asks whether an arbitrary finite group G can be realized as Gal(K/Q) for some Galois extension K of Q. When such a realization has been given for a particular G then a natural sequel is to find arithmetical realizations of the irreducible representations of G. One possibility is to ask for realizations in the Mordell-Weil groups of elliptic curves over Q: Given an irreducible complex representation τ of Gal(K/Q), does there exist an elliptic curve E over Q such that τ occurs in the natural representation of Gal(K/Q) on C ⊗Z E(K)? The present paper does not attempt to investigate this question directly. Instead we adopt Greenberg’s point of view in his remarks on nonabelian Iwasawa theory [5] and consider a related question about root numbers. Let ρE denote the representation of Gal(K/Q) on C ⊗Z E(K) and 〈τ, ρE〉 the multiplicity of τ in ρE , and write L(E, τ, s) for the tensor product L-function associated to E and τ . The conjectures of Birch-Swinnerton-Dyer and Deligne-Gross imply that