Extendible Functions and Local Root Numbers Remarks on a paper of R.P. Langlands

TL;DR

This paper investigates the algebraic conditions under which local root numbers and epsilon-factors can be extended to all virtual representations of Weil groups, focusing on the algebraic framework within solvable profinite groups.

Contribution

It provides a complete algebraic analysis of the relations needed for the extension of local root numbers, using modifications of Brauer's theorem within the context of solvable profinite groups.

Findings

Established a set of minimal relations for extendible functions

Demonstrated the algebraic criteria for the existence of epsilon-factors

Connected the algebraic framework to local root number extensions

Abstract

This paper refers to Langlands' big set of notes [L] devoted to the question if the (normalized) local Hecke-Tate root number $\Delta=\Delta(E,\chi)$, where $E$ is a finite separable extension of a fixed non-archimedean local field $F$, and $\chi$ a quasicharacter of $E^\times$, can be appropriately extended to a local $\varepsilon$-factor $\varepsilon_\Delta=\varepsilon_\Delta(E,\rho)$ for all virtual representations $\rho$ of the corresponding Weil group $W_E.$ Whereas Deligne [D] has given a relatively short proof by using the global Artin-Weil L-functions, the proof of Langlands is purely local and splits into two parts: the {\bf algebraic part} to find a minimal set of relations for the functions $\Delta$, such that the existence (and uniqueness) of $\varepsilon_\Delta$ will follow from these relations; and the more extensive {\bf arithmetic part} to give a direct proof that all these relations are actually fulfilled. Our aim is to cover the algebraic part of Langlands' notes which can be done completely in the framework of representations of solvable profinite groups, where two modifications of Brauer's theorem play a prominent role. Introduction / 1. The notion of extendible functions / 2. The kernel of the Brauer map and its generating relations / 3. A criterion for the extendibility of functions / 4. Recovering Theorem 3.1 for the case of local root numbers / Appendices A1. Proving Brauer 3 and Brauer 4 / A2. On type-III-groups.