On liftings and cusp cohomology of arithmetic groups

Abstract

w 0. IntroductionLet Fc G(E) be an arithmetic subgroup of a semi simple connected algebraic group G defined over an algebraic number field E, and let V be a finite dimensional complex representation of the group G~ = G(N| of real points of G. Then the cusp cohomology it/cusp(/", V) is a subspace of the cohomology of F with coefficients in V defined in analytical terms [3]; using the identification of the cohomology H*(F, V) of F with the relative Lie algebra cohomology it is given as Hc*usp(F, V) = H*(g, Ko~; L20(F\G 0o) | V) (1) and, in particular, it isolates a finite set (depending on V) of representations n of G o occurring in the cuspidal spectrum L20(F\G~) with finite multiplicities re(n, F). The study of the cusp cohomology of F is therefore the study of special types of cuspidal automorphic forms.For example, if G=SL2/~, FcSL2(~. ) and V= V k is of dimension k, then (1) is the Eichler-Shimura isomorphism 1 H~usp(r, Vk) ~-S~+ 1 (r) | s~+ , (r)where S~+I(F ) is the space of holomorphic or anti-holomorphic cusp forms with respect to F of weight k+ 1; and the dimension of S~+ I(F) is the multiplicity m(D~,F) of the holomorphic (antiholomorphic) discrete series representation D~ of SL2(~ ). More generally, if G~o has discrete series representations the Selberg trace formula allows one to get hold of cusp cohomology classes by computing Euler-Poincar6 characteristics (cf. w But if G~ has no discrete series (e.g. in case G =SL, with n > 2 or G~--H(~) is a complex Lie group) the Euler-Poincar6 characteristics of the representations occurring in (1) vanish and the ordinary trace formula does not seem to be of any direct help.On the other hand, there are geometrical methods as additional tools. For example, if F~SL2((DE), (~s a ring of imaginary quadratic integers, or F

Functoriality and Special Values of L-Functions

“…This theorem is essentially due to Labesse and Schwermer [26]. We then digress a little and discuss the issue of functoriality and a representation being cohomological.…”

“…We end the section by a series of examples illustrating some of these principles. But before doing so, we would like to draw the reader's attention to a conjecture of Clozel [7, §1] which is motivated by the ideas of Labesse and Schwermer [26]. It roughly states that given a tempered cohomological representation at infinity, one can find a global cuspidal automorphic representation whose representation at infinity is the given one.…”

“…(1) Labesse and Schwermer [26] proved the existence of nontrivial cuspidal cohomology classes for SL 2 and SL 3 over any number field E which contains a totally real number field F such that F = F 0 ⊂ F 1 ⊂ · · · ⊂ F n = E with each each F i+1 /F i either a cyclic extension of prime degree or a non-normal cubic extension. The functorial lifts used were base change for GL 2 and the symmetric square lifting of Gelbart and Jacquet.…”

“…(2) Motivated by [26], Clozel [7] used automorphic induction and proved the existence of nontrivial cuspidal cohomology classes for SL 2n over any number field. (3) Rajan [35], also motivated by [26], proved the existence of nontrivial cuspidal cohomology classes for SL 1 (D) for a quaternion division algebra D over a number field E, with E being an extension of a totally real number field F with solvable Galois closure. Other than base change, he used the JacquetLanglands correspondence.…”

Surface subgroups of Kleinian groups with torsion

“…In all the cases where the arithmetic methods of [10] and [5] could produce a finite index subgroup with positive first Betti number, the author, Long and Reid in [14] were able to show that Γ is in fact large.…”