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

Sazzad Ali Biswas

TL;DR

This paper extends Deligne’s twisting formula for local root numbers to the case of minimal conductor U-isotropic Heisenberg representations of dimension prime to $p$, and simultaneously generalizes the Deligne–Henniart twisting framework to broader zero-dimensional settings. By explicitly computing Artin and Swan conductors and exploiting decompositions into tame, unramified, and $p$-power components, it derives a concrete twisting formula for $W(\sigma\otimes\rho,\psi)$ when $\rho$ is a U-isotropic Heisenberg representation and under appropriate conductor/jump conditions. Two key applications follow: (i) an invariant root-number formula for Heisenberg representations, expressing $W(\rho,\psi)$ in terms of Langlands data and the relevant characters, and (ii) a local converse theorem on the Galois side, stating that equality of twisted root numbers with a fixed $\rho_m$ forces $\rho_1$ and $\rho_2$ to be equivalent up to an unramified twist. The results illuminate how local root numbers behave under twisting for a rich class of Galois representations and yield concrete criteria for Galois-side classifications.

Abstract

Let $F/\bbQ_p$ be a non-Archimedean local field, and $G_F$ be the absolute Galois group of $F$. Let $ρ_1$ and $ρ_2$ be two finite-dimensional complex representations of $G_F$. Let $ψ$ be a nontrivial additive character of $F$. Then, the question is: What is the twisting formula for the root number $W(ρ_1\otimesρ_2,ψ)$?} In general, the answer to this question is not yet known. However, if one of $ρ_i \quad(i=1,2)$ is one-dimensional with ``sufficiently'' large conductor, then in [13], Deligne gave a twisting formula for $W(ρ_1\otimesρ_2,ψ)$. 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.

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

TL;DR

This paper extends Deligne’s twisting formula for local root numbers to the case of minimal conductor U-isotropic Heisenberg representations of dimension prime to , and simultaneously generalizes the Deligne–Henniart twisting framework to broader zero-dimensional settings. By explicitly computing Artin and Swan conductors and exploiting decompositions into tame, unramified, and -power components, it derives a concrete twisting formula for when is a U-isotropic Heisenberg representation and under appropriate conductor/jump conditions. Two key applications follow: (i) an invariant root-number formula for Heisenberg representations, expressing in terms of Langlands data and the relevant characters, and (ii) a local converse theorem on the Galois side, stating that equality of twisted root numbers with a fixed forces and to be equivalent up to an unramified twist. The results illuminate how local root numbers behave under twisting for a rich class of Galois representations and yield concrete criteria for Galois-side classifications.

Abstract

Let be a non-Archimedean local field, and be the absolute Galois group of . Let and be two finite-dimensional complex representations of . Let be a nontrivial additive character of . Then, the question is: What is the twisting formula for the root number ?} In general, the answer to this question is not yet known. However, if one of is one-dimensional with ``sufficiently'' large conductor, then in [13], Deligne gave a twisting formula for . 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 . In this paper, we first extend Deligne's twisting formula for U-isotropic Heisenberg representation of dimension prime , 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.

Paper Structure

This paper contains 12 sections, 26 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Ramification break
  4. Heisenberg Representations
  5. U-isotropic Heisenberg representations
  6. Arithmetic Description of Heisenberg Representations
  7. Computation of the Artin and Swan conductors, and the dimension theorem
  8. Proof of Theorems \ref{['Theorem using Deligne-Henniart']} and \ref{['General twisting formula by minimal Heisenberg representation']}
  9. Applications
  10. Invariant root number formula for Heisenberg representations
  11. Converse theorem on the Galois side
  12. Appendix

Key Result

Theorem 1.1

Let $\rho$ be a virtual representation of $G_F$ (without moderate component). There exists an element $\gamma\in F^\times$ uniquely determined modulo $U_F^{j(\rho)/2-1}$, such that for any virtual representation $\rho_0$ of $G_F$ of dimension zero which satisfies $\beta(\rho_0)<j(\rho)/2$, we have Here $\nu_F(\gamma)=a(\rho)+\dim(\rho)\cdot n(\psi)$, and $j(\rho)$ is the jump of $\rho$. $\beta(\r

Theorems & Definitions (64)

  • Theorem 1.1: Deligne-Henniart's Twisting formula, Theorem 4.6, DH
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Invariant Formula
  • Theorem 1.5: Converse Theorem on the Galois side
  • Definition 2.1: Artin and Swan conductors
  • Definition 2.2: Jump for a representation
  • Theorem 3.1: Zink, Z2, p. 301, Corollary 1.2
  • Remark 3.2
  • Lemma 3.3
  • ...and 54 more