Local root numbers of two-dimensional symplectic respresentations

Abstract

Let F be a non-archimedean local field with residue field F_q and q odd.  Consider A the unique quaternion division algebra over F.  We prove the existence of a homomorphism of the form Γ̂;_F : RO(A*/F*) → μ₄ = {±1, ±i} analogous to Γ_F : RO(F) → μ₄ given in [31].  Using Γ̂;_F and the results of D.  Prasad and D. Ramakrishnan [22] regarding the Langlands correspondence, we construct γ_F, a map from two-dimensional symplectic Galois representations to fourth roots of unity. If σ is a two-dimensional symplectic Galois representation, this construction, when q ≡ 1(mod.4), gives a formula for the local root number of σ in terms of γ_F(σ).

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