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\(p\)-adic \(L\)-functions and unitary completions of representations of \(p\)-adic reductive groups. (English) Zbl 1092.11024

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.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F85 \(p\)-adic theory, local fields
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
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