Mathematical Research Letters
Volume 23 (2016)
Interpolating local constants in families
Pages: 1789 – 1817
We extend the theory of local constants developed by Jacquet, Piatetski–Shapiro, and Shalika in  to $\ell$-adic families of representations of $GL_n(F)$ where $F$ is a $p$-adic field with $\ell$ not equal to $p$. We construct zeta integrals and gamma factors for representations coming from the conjectural “local Langlands correspondence in families” of Emerton–Helm, proving their rationality and functional equation. We also construct a universal gamma factor with coefficients in the integral Bernstein center.