Banach space representations and Iwasawa theory

By P. Schneider, Jeremy Teitelbaum. 2000

170 citations recorded by the source corpus.

We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringoK[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗oK[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL2(ℤp).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8ca17c4f093d3fa074056783579c63d9694cbaee

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Citations in the corpus

Algebras of p-adic distributions and admissible representations (2002, methodology): “ly analytic G-representation, and we obtain Van = V 1 as claimed. For this continuity it suﬃces to check that the strong topology on (V′ bs) ′is ﬁner than the Banach space topology on V. According to [ST3] the Banach space V is identiﬁed with Homcont o (M,K) with norm given by the sup-norm, and the ”
The p-adic local Langlands correspondence for GL_2(Q_p) (2013, background): “In particular, the image of the morphism X → Π is closed [57].”
The emerging p-adic Langlands programme (2011, background): “Strangely, the terminology “p-adic Langlands correspondence/programme” started to spread (at least in the author’s memory) only shortly after preprints of [6], [7], [8], [9], [14], [19], [24], [53], [59], [60], [61], [70] were available (that is, around 2004), although of course p-adic consideration”
On irreducible representations of compact $p$-adic analytic groups (2011, background): “In a series of papers including [68, 69, 70, 71] Schneider and Teitelbaum study a class of rings that they call the distribution algebras D(G,K) of a p-adic analytic group G.”
Ring-Theoretic Properties of Iwasawa Algebras: A Survey 1 (2005, background): “2000 Mathematics Subject Classification: 16L30, 16P40, 20C07, 11R23 Keywords and Phrases: Iwasawa algebra; compact p-adic analytic group; complete noetherian semilocal ring; Auslander-Gorenstein condition”
The image of Colmez’s Montreal functor (2010, methodology): “We note that it follows from [58] and [29, 6.”
On the structure theory of the Iwasawa algebra of a p-adic Lie group (2001, background): “Schneider and Teitelbaum proved in [32] that, for a compact p-adic Lie group, these categories are anti-equivalent to the category of certain Banach space representations of G.”
Patching and the p-adic local Langlands correspondence (2013, background): “A is a linear topological O-module, we write A∨ for its Pontrjagin dual Homcont O (A,E/O), where E/Ohas the discrete topology, and we give A∨ the compact open topology. By the proof of Theorem 1.2 of [ST02], the functor A→Ad:= Homcont O (A,O) induces an anti-equivalence of categories between the cat”
Locally analytic vectors in representations of locally p-adic analytic groups (2004, background): “G-representation (introduced in [20]) and also the notion of admissible continuous G-representation (introduced in [ 22 ]).”
On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms (2006, methodology): “(Equivalently, a compact p-adic analytic group, in the terminology of [25].) We fix a finite extension E of Q p, with ring of integers OE. Recall that the category of admissible continuous representations of G over E (as defined and studied in [ 30 ] and [13, §6.2]; we employ the notation of the lat”
ORDINARY PARTS OF ADMISSIBLE REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPS I. DEFINITION AND FIRST PROPERTIES (2007, methodology): “In this subsection we introduce a class of representations related to the considerations of [15].”
On the Breuil–Mézard conjecture (2012, background): “equipped with the supremum norm is an admissible unitary L-Banach space representation of G ×G, see [61].”
The eigencurve over the boundary of weight space (2014, methodology): “s: its dual is the coinvariant subspace of Emerton’s completed homology under the action of the unipotent radical of the Borel subgroup at p, which is a compact topological -module (in the 5 sense of [ST02]). In this paper, we present the construction using induced representations; this gives rise t”
A Local-Global Compatibility Conjecture in the p-adic Langlands Programme for GL 2/Q (2006, background): “We let T̂ denote the rigid analytic space over E that classifies the continuous (equivalently, locally analytic) characters of T(Qp).”