Galois theory, elliptic curves, and root numbers

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