Interpolating gamma factors in families
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In this thesis, we extend the results of Jacquet, Piatetski-Shapiro, and Shalika [JPSS83] to construct interpolated local zeta integrals and gamma factors attached to families of admissible generic representations of GL[subscript n](F) where F is a p-adic field. Our families are parametrized by the spectrum of an ℓ-adic coefficient ring where ℓǂp. To show the importance of gamma factors, we prove a converse theorem in families, which says that suitable collections of interpolated gamma factors of pairs uniquely determine a family of representations, up to supercuspidal support. To prove the converse theorem we re-prove a classical vanishing Lemma, originally due to Jacquet and Shalika, in the setting of families. This is done by extending the geometric methods of Bushnell and Henniart to families, via Helm's theory of the integral Bernstein center.