## $$p$$-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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