\(p\)-adic \(L\)-functions and unitary completions of representations of \(p\)-adic reductive groups
type: Item
instance of: scholarly article
title: \(p\)-adic \(L\)-functions and unitary completions of representations of \(p\)-adic reductive groups
zbMATH Open document ID: 1092.11024
author: Matthew Emerton
published in: Duke Mathematical Journal
publication date: 2006-03-10T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.dmj/1132064630
review text: Let \(G\) be a reductive \(p\)-adic group. Let \(V\) be a locally convex \(p\)-adic vector space supplied with a continuous action of \(G\). A ``universal unitary completion'' of \(V\) is a solution \(\widehat{V}\) of the universal problem posed by the \(G\)-equivariant continuous linear maps \(V\rightarrow B\), where \(B\) is a \(p\)-adic Banach space with a unitary action of \(G\). The author developes some general theory on this notion, giving necessary or sufficient conditions for \(\widehat{V}\) to exist or, when it exists, to be nonzero. In the greater part of the paper we have \(G=\text{GL}(2,{\mathbb Q}_p)\). For parabolically induced representations of \(G=\text{GL}(2,{\mathbb Q}_p)\) necessary conditions are derived for the existence of a nonzero completion. That these conditions are also sufficient is proved, in specified cases where the local representation comes from a newform, by embedding the local representation into the completed \(p\)-adic cohomology of a modular curve. The techniques of this article are applied to give a new definition for \(p\)-adic \(L\)-functions and a proof of the Mazur-Tate-Teitelbaum conjecture on extra zeros with Breuil's \({\mathcal L}\)-invariant.
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 46S10
zbMATH DE Number: 5011513
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5011513