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01 November 05 P -adic L -functions and unitary completions of representations of P -adic reductive groups
Matthew Emerton
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Duke Math. J. 130(2): 353-392 (01 November 05). DOI: 10.1215/00127094-8230018


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 GL 2 , and we relate our construction to certain conjectures of Breuil in [4, Sec. 1.3] for the group GL 2 ( 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 GL 2 ( 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 GL 2 ( 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 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 L -invariant)


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Matthew Emerton. " P -adic L -functions and unitary completions of representations of P -adic reductive groups." Duke Math. J. 130 (2) 353 - 392, 01 November 05.


Published: 01 November 05
First available in Project Euclid: 15 November 2005

zbMATH: 1092.11024
MathSciNet: MR2181093
Digital Object Identifier: 10.1215/00127094-8230018

Primary: 11F67
Secondary: 11F70 , 11F85 , 22E50 , ‎46S10

Rights: Copyright © 2005 Duke University Press


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Vol.130 • No. 2 • 01 November 05
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