Derived Langlands

This appendix contains an account of a calculation, by Deligne and Henniart, of wildly ramified local roots numbers modulo roots of unity. Since this result is relevant to epsilon factors derived from monomial resolutions of GLn of a local field I have included the account which has been gathering dust on my computer since 2010 or earlier and on my homepage since 2012. This is the homepage version reproduced “as is” here we go! Originally, in other lectures in the series which begat this Appendix, one encounters local L-functions, functional equations and local epsilon factors of admissible representations. The p-adic Galois epsilon factors are numbers lying on the unit circle and they are fundamental in the local Langlands correspondence which was proved by Mike Harris and Richard Taylor. Later part of the proof was simplified by Henniart using his “uniqueness theorem”, which is characterised in terms of p-adic epsilon factors. This Appendix, which was formerly a lecture in the above mentioned series, is mainly expository. In it I shall outline the calculation by Deligne and Henniart of the p-adic epsilon factors of wild, homogeneous Galois representations modulo p-primary roots of unity. This formula is an important ingredient in the proofs of the uniqueness theorem. The only novel ingredients in my exposition will be the use of monomial resolutions to reduce to the one-dimensional case and an explicit formulae for the Deligne-Henniart “Gauss sum” (which seems in my opinion to contradict, in the tamely ramified case, one of the lemmas claimed in general but used by Henniart only in the wild case at the crux of the proof). 1. The basic ingredients 1.1. These notes are an exposition of the papers [49] and [73] which culminate in the derivation of a formula for Galois local constants (otherwise known as Galois epsilon factors) modulo p-power roots of unity for wildly ramified, homogeneous representations on the Weil group of a p-adic local field. My account will differ from [49] in §3.3, which I shall derive using monomial resolutions. Furthermore, since it is well-known how to pass