Gamma Factors of Pairs and a Local Converse Theorem in Families

We prove a $GL(n)\times GL(n-1)$ local converse theorem for $\ell $-adic families of smooth representations of $GL_n(F)$ where $F$ is a finite extension of ${\BBQ}_p$ and $\ell \neq p$. Along the way, we extend the theory of Rankin–Selberg integrals, first introduced in Jacquet et al. [20], to the setting of families, continuing previous work of the author [23].