Endoscopy and cohomology growth on \(\mathrm{U}(3)\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1315.11043
DOI: 10.1112/S0010437X13007720
author: Simon Marshall
published in: Compositio Mathematica
publication date: 2014-08-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.7244
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 6333836
zbMATH Keywords: arithmetic groups
zbMATH Keywords: cohomology
zbMATH Keywords: automorphic forms
zbMATH Keywords: endoscopy
MaRDI profile type: Publication
arXiv ID: 1301.7244
cites work: CYCLIC ODD DEGREE BASE CHANGE LIFTING FOR UNITARY GROUPS IN THREE VARIABLES
cites work: The orders of the classical simple groups
cites work: Bounds for multiplicities of automorphic representations
cites work: The index theorem and the trace formula
cites work: Fake projective planes
cites work: Äquivalenz Hermitescher Formen über einem beliebigen algebraischen Zahlkörper
cites work: \(L\)-functions and Fourier-Jacobi coefficients for the unitary group \(U(3)\)
cites work: Characters, genericity, and multiplicity one for \(U(3)\)
OpenAlex ID: W3104140333
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title: Endoscopy and cohomology growth on \(\mathrm{U}(3)\)
review text: Let \(\mathrm{U}(2,1,{\mathbb R})\) be the isometry group of an Hermitian form in three variables of signature \((2,1)\). A way to construct cocompact discrete subgroups of \(\mathrm{U}(2,1,{\mathbb R})\) is as follows: let \(E/F\) be a CM-extension of number fields, that is, \(F\) is a totally real number field and \(E\) is a quadratic extension without real places (i.e., at each infinite place of \(F\) the extension \(E/F\) is ramified). Let \(\Phi\) be an Hermitian form on \(E^3\) such that \(\Phi\) is of signature \((2,1)\) at exactly one infinite place \(v_0\) of \(F\) and definite at all other infinite places. Let \(\mathcal O\) denote the ring of integers in \(E\) and \(\Gamma\) be the stabilizer of the lattice \({\mathcal O}^3\) in \(U(\Phi)\); then \(\Gamma\) is a lattice in the Lie group \(G = \mathrm{U}(\Phi\otimes_{v_0} {\mathbb C})\cong \mathrm{U}(2,1)\). If \(F\not= {\mathbb Q}\) then \(\Gamma\) is actually cocompact (when \(E\) is an imaginary quadratic field the lattice \(\Gamma\) is never cocompact). The group \(\Gamma\) acts cocompactly on the contractible manifold \(\mathrm{U}(2,1)/(\mathrm{U}(2)\times \mathrm{U}(1))\), hence its virtual cohomological dimension is 4 and any of its torsion-free finite index subgroups is a Poincaré duality group of dimension 4.NEWLINENEWLINEIf \(\mathfrak n\) is an ideal in \(\mathcal O\) then one can define a congruence subgroup of level \(\mathfrak n\) in \(\Gamma\), which we denote by \(\Gamma({\mathfrak n})\). This is a finite index subgroup of \(\Gamma\) (of index \(V({\mathfrak n}) := [\Gamma:\Gamma({\mathfrak n})] \asymp |\mathfrak n|^8\)). The subject of the present paper is the growth rate of the cohomology groups \(H^i(\Gamma({\mathfrak n}, {\mathbb C})\) relatively to the index \(V({\mathfrak n})\). We denote by \(b^i(\Gamma(\mathfrak n)) = \dim H^i(\Gamma({\mathfrak n}, {\mathbb C})\). Then it is known by work of \textit{D. L. DeGeorge} and \textit{N. R. Wallach} [Ann. Math. (2) 107, 133--150 (1978; Zbl …
reviewed by: Jean Raimbault
description: scientific article; zbMATH DE number 6333836