An extension of Deligne-Henniart's twisting formula and its applications

Abstract

Let $F/{\mathbb{Q}}_p$ be a non-Archimedean local field, and $G_F$ be the absolute Galois group of $F$. Let ${\rho }_1$ and ${\rho }_2$ be two finite-dimensional complex representations of $G_F$. Let $\psi$ be a nontrivial additive character of $F$. Then, the question is: What is the twisting formula for the root number $W({\rho }_1\otimes {\rho }_2,\psi )$?} In general, the answer to this question is not yet known. However, if one of ${\rho }_i(i=1,2)$ is one-dimensional with ``sufficiently'' large conductor, then in [13], Deligne gave a twisting formula for $W({\rho }_1\otimes {\rho }_2,\psi )$. Later, in [12], Deligne and Henniart gave a general twisting formula for a {\it zero}-dimensional virtual representation twisted by a finite-dimensional representation of $G_F$. In this paper, we first extend Deligne's twisting formula for U-isotropic Heisenberg representation of dimension prime $p$, then we further extend Deligne-Henniart's result. Finally, we provide two very important applications of our twisting formula: -- (i) invariant formula for the local root numbers for U-isotropic Heisenberg representations, and (ii) a converse theorem on the Galois side.

References in corpus (2)

• Towards the Jacquet Conjecture on the Local Converse Problem for $p$-adic ${\mathrm{GL}}_n$
• Lamprecht-Tate Formula