\(p\)-adic \(L\)-functions and unitary completions of representations of \(p\)-adic reductive groups
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.
scientific article; zbMATH DE number 5011513
scientific article