P -adic L -functions and unitary completions of representations of P -adic reductive groups

Abstract

In the first half of the article, we introduce the notion of the universal unitary completion of a continuous representation of a $p$-adic reductive group on a locally convex $p$-adic vector space, and we prove that such a completion exists under appropriate hypotheses. The problem of studying unitary completions has been raised by Breuil [4] in connection with his work on a possible $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2$, and we relate our construction to certain conjectures of Breuil in [4, Sec. 1.3] for the group ${\mathrm{GL}}_2({\mathbb{Q}}_p)$. In particular, we show that the universal unitary completion of the locally analytic parabolic induction of a locally algebraic character coincides with the universal unitary completion of the corresponding locally algebraic induction, provided that the character being induced satisfies a noncritical slope condition (see Prop. 2.5). In the second half of the article, we consider a certain unitary Banach space representation of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$ obtained by $p$-adically completing the cohomology of classical modular curves. The mere existence of this representation implies that those locally algebraic, parabolically induced representations of ${\mathrm{GL}}_2({\mathbb{Q}}_p)$ which arise from classical finite-slope newforms have a nontrivial universal unitary completion (verifying a conjecture of Breuil in [4, Sec. 1.3] for these representations), while applying Proposition 2.5 in this context enables us to give a new construction of $p$-adic $L$-functions attached to $p$-stabilized newforms of noncritical slope. Combining our construction with the representation-theoretic definition of $\mathcal{L}$-invariants implicit in [5, Cor. 1.1.7], we are able to give a simple proof of the Mazur-Tate-Teitelbaum exceptional zero conjecture in [18, p. 46] (in terms of Breuil's definition of the $\mathcal{L}$-invariant)