The Langlands conjecture for \(\mathrm{GL}_2\) of a local field
type: Item
instance of: scholarly article
title: The Langlands conjecture for \(\mathrm{GL}_2\) of a local field
zbMATH Open document ID: 0469.22013
DOI: 10.2307/1971151
published in: Annals of Mathematics. Second Series
publication date: 1980-01-01T00:00:00Z
review text: Let \({\mathbf K}\) be a field and \(k\) be a positive integer. The set of elements in \({\mathbf K}\) that can be written as sums of \(k\)th powers (resp.\ as sums of \(k\)th powers of totally positive elements) shall be denoted by \(P({\mathbf K},k)\) (resp.\ \(P^+({\mathbf K},k)\)). \(w({\mathbf K},k)\) (resp.\ \(g({\mathbf K},k)\)) then denotes the smallest positive integer \(n\) such that each element in \(P({\mathbf K},k)\) (resp.\ \(P^+({\mathbf K},k)\)) can be written as a sum of \(n\) \(k\)th powers (resp.\ as a sum of \(n\) \(k\)th powers of totally positive elements), provided such an integer exists. Otherwise, one puts \(w({\mathbf K},k)=\infty\) (resp.\ \(g({\mathbf K},k)=\infty)\). Recall that by Artin-Schreier theory, the totally positive elements are exactly the nonzero elements in \(P({\mathbf K},2)\). In the literature, \(w({\mathbf K},2)\) is often called the Pythagoras number of \({\mathbf K}\). This paper contains many nice results regarding the invariants \(w\) and \(g\), proved by elementary methods. The paper is mainly based on the author's PhD thesis from 1970, and having the results available now in such published form is certainly highly appreciated by all those working on these and related problems. Here are some of the main results. One always has \(P^+({\mathbf K},k)=P({\mathbf K},2k)\), and for any odd \(k\) or any non formally real \({\mathbf K}\) one has \({\mathbf K}=P({\mathbf K},k)\) and \(w({\mathbf K},k)<\infty\). If \({\mathbf K}\) is not formally real of characteristic \(0\), then furthermore \(w({\mathbf K},k)=g({\mathbf K},k)<\infty\) for all positive \(k\). Let \(\alpha\in P({\mathbf K},k)\) and define \(\ell_k(\alpha)\) (resp. \(\ell_k^+(\alpha)\)) to be the smallest integer \(n\) such that \(\alpha\) is a sum of \(n\) \(k\)th powers (resp.\ of \(n\) \(k\)th powers of totally positive elements). If \({\mathbf K}\) is formally real and \(\alpha\in P({\mathbf K},2)\), the author shows that there exists a positive integer \(\…
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11S37
zbMATH DE Number: 3736082
zbMATH Keywords: 2-dimensional semi-simple continuous complex representations
zbMATH Keywords: absolute Weil group
zbMATH Keywords: irreducible admissible non-special representations
zbMATH Keywords: central character
zbMATH Keywords: supercuspidal representations
Wikidata QID: Q55967375
author: Philip C. Kutzko
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971151
OpenAlex ID: W2314665678
description: scientific article; zbMATH DE number 3736082