# Overview

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## Arithmetic quotients of locally symmetric spaces and their cohomology

**The presentations videos are now available here:http://www.crm.umontreal.ca/2020/Arithmetic20/horaire_e.html
as well as on the CRM YouTube page:https://www.youtube.com/watch?v=ILIsBJigo58&list=PLHaWeIntAtAL739Snl3LsAYamxHi6NNkm**

If G is a reductive algebraic group over Z, the group G(Z) of its integral points (or any congruence subgroup thereof) acts discretely on the locally symmetric space X:= G(R)/K, where K is a maximal compact subgroup of G(R). The quotients G(Z) X play a fundamental role in the theory of automorphic forms and in number theory. Notably, their cohomology is a rich source of invariants attached to automorphic representations of G, and thus plays a central role in the Langlands program. A fundamental trichotomy governing the topological behaviour of such arithmetic quotients was proposed around 2010 by Bergeron and Venkatesh. A single positive integer d, depending only on the overlying symmetric space X, dictates the expected behaviour of the homology of the arithmetic quotient. When d=0, the cohomology is expect to have very little torsion but lots of characteristic 0 homology, which can be studied via analytic and transcendental methods (de Rham cohomology, Hodge theory). Shimura varieties and even-dimensional real hyperbolic spaces fall into this class. When d=1, one expects to find a lot of torsion but very little characteristic 0 homology. Odd dimensional hyperbolic manifolds, such as the Bianchi three-fold SL2(Z[i]) SL2(C)/U(2), fall into this case. When d is greater than 1, one expects little torsion and little characteristic zero homology.

There has been remarkable progress towards understanding how this trichotomy interacts with arithmetic: When d = 0, several interesting recent torsion-freeness results have been obtained by researchers like Caraiani, Emerton, Gee, and Scholze. When d=1, one can ask whether torsion always arises when it's expected to, and with the expected abundance. Torsion can be probed analytically using the Cheeger-Muller theorem. But there are obstructions ("tiny eigenvalues" and "very complex cycles"), which are very interesting in their own right, and need to be overcome in order to prove that there's as much torsion as expected. This torsion growth problem, especially for hyperbolic three-manifolds, has a life of its own even outside number theory, notably in the community of geometric groups theorists. Among the most striking developments arising in the relatively less well explored setting where d is larger than 1, let us mention Peter Scholze's construction of Galois representations attached to (possibly torsion) eigenclasses in the cohomology of arithmetic quotients, which is especially deep in this case. Another highly promising, fundamental breakthrough is manifested in Akshay Venkatesh's conjecture on derived Hecke algebras, which is expected to play an important role in extending the scope of the Taylor-Wiles method beyond the setting of d=0 to which it had been confined until relatively recently. The deep study of torsion in homology and analytic torsion carried out earlier by Bergeron, Venkatesh and others played a very important part in the nascent theory of derived Hecke operators and the attendant motivic action on the cohomology of arithmetic groups. In some very special instances, where G=GL(2) and one focusses on the coherent cohomology of an arithmetic quotient with values in certain automorphic sheaves, Venkatesh's conjectures exhibit a tantalising connection with certain ``tame refinements", in the spirit of conjectures of Mazur and Tate, of conjectures on the values of triple product p-adic L-functions.

The field is still in a very exploratory stage in which precise expectations (conjectural or otherwise) have not yet fully cristallised. For instance, there does not yet seem to be a reasonable conjecture about "how much cohomology", torsion or characteristic zero, to expect when d is greater than 1. Among other reasons, this makes computing in this setting very interesting. The workshop is expected to have a significant computational and experimental component, in which various experts will report on experimental data that might prove valuable in solidifying our expectations.