On Emerton’s p-adic Banach Spaces

Abstract

Abstract. The purpose of the current paper is to introduce some new methods for studying the p-adic Banach spaces introduced by Emerton [9]. We first relate these spaces to more familiar sheaf cohomology groups. As an application, we obtain a more general version of Emerton's spectral sequence. We also calculate the spaces in some easy cases. As a consequence, we obtain a number of vanishing theorems. 1.1.Cohomology of arithmetic quotients. Let G be a linear algebraic group over a number field k. We choose a maximal compact subgroup K ∞ ⊂ G(k ∞ ) and let K • ∞ be the identity component in K ∞ . This paper is concerned with the cohomology of the following "arithmetic quotients":Fix once and for all a finite prime p of k and let p be the rational prime below p. By a "tame level" we shall mean a compact open subgroup K p of G(A p f ). For a field E of characteristic zero, the level K p Hecke algebra is defined by H(K p , E) = {f : K p \G(A p f )/K p → E : f has compact support}. Given a finite dimensional algebraic representation W of G over an extension E/k p , one defines a local system V W on each arithmetic quotient Y (K f ). We define the classical cohomology groups of tame level K p as follows: where K p ranges over the compact open subgroups of G(k p ). The symbol " * " denotes either the empty symbol, meaning usual cohomology, or "c", meaning compactly supported cohomology. There is a smooth action of G(k p ) on H • cl. (K p , W ), and there is also an action of the level K p Hecke algebra H(K p , E). The systems of Hecke eigenvalues arising in H • cl., * (K p , W ) are of considerable interest in number theory. Traditionally, the classical cohomology groups have been studied using the theory of automorphic representations (for example in [4]). Recently, Emerton introduced a new method for studying the classical cohomology groups. Instead of studying the space of automorphic forms on G, Emerton introduced a collection of p-adic Banach spaces, from which the classical cohomology can be recovered. Emerton's spaces are defined as follows:For a finite extension E/Q p we also definẽhas the structure of a Banach space over E, where the unit ball is defined to be the O E -span of the image ofH • * (K p , Z p ). It is also convenient to consider the direct limit of these groups over the tame levels:We have the following actions on these spaces:(1) The group G(A p f ) acts smoothly onH • * (G, Z p ); the subspaceH • * (K p , E) may be recovered as the K p -invariants inH • * (G, E). (2) The Hecke algebra H(K p , E) acts onH • * (K p , E). (3) The group G(k p ) acts continuously, but not usually smoothly on the spacesH • * (K p , −). (4) For a finite extension E/k p , we defineH • * (K p , E) kp−loc.an. to be the subspace of k p -locally analytic vectors inH • (K p , E) (see [10]). The Lie algebra g of G acts on the subspaceH • * (K p , E) kp−loc.an. . Emerton proved (Theorem 2.2.11 of [9]) that his spaces are related to the classical cohomology groups by the following spectral sequence of smoothThe spectral sequenc...

Completed cohomology – a survey

“…Conjecture 6.1 implies Hill's conjecture, but is quite a bit stronger in general. The various vanishing results proved in [6, §5], being consistent with [6,Conj. 3], are thus also consistent with our general conjecture.…”

“…Richard Hill has also made a conjecture about the vanishing of completed cohomology of arithmetic quotients, namely [6,Conj. 3].…”

“…This construction was first introduced in paper [4] (although insufficient attention was given there to the integral aspects of the theory), and then further developed in the papers [2] and [6]. The papers [4] and [2] may give the impression that p-adically completed cohomology is some sort of auxiliary construction that can be used to prove theorems (of either a p-adic or classical nature) about automorphic forms.…”

Vanishing theorems for Shimura varieties at unipotent level

“…We note that the existence of such a spectral sequence has been previously announced by Emerton, relying on his theory of ordinary parts [Eme10a,Eme10b]. In this paper, we give a different construction relying on computations of (co)homology using singular and simplicial chains, in the style of Ash-Stevens [AS07], as well as ideas of Hill [Hil10]. 5.1.…”

Infinitesimal characters in arithmetic families

“…Proof. We use an argument due to Hill [47] in an easier setting, alternatively see [32, (2.1.10)]. We pick a triangulation of Y (K p f K p ) and write down the Čech complex computing the cohomology of the local system associated to C(K p /Λ p , O/̟ s ).…”