These conditions form a subtle combination of geometric conditions, arising from cohomological considerations, and arithmetic conditions, arising from the analytic properties of Eisenstein series and given in terms of automorphic L -functions. This paper is a survey of the most important results of our long-lasting collaboration. In this note we discuss a special case of a general theme: How do special values of L -functions influence the structure of the cohomology of arithmetic groups?
[PDF] Globally maximal arithmetic groups - Semantic Scholar
We study whether the property of being cohomological is preserved under Langlands functoriality for the transfer of tempered representations from real classical Lie groups to an appropriate general linear group. We survey known results, as well as including some new results that make partial progress on the conjecture. Let F be a field of characteristic 0 containing all roots of unity. The construction is based on the ring of rational Witt vectors of F. We also discuss a variant of this construction when the field is not required to contain all roots of unity, in which case there are natural Frobenius-type automorphisms which encode the descent along the cyclotomic extension.
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Deligne and Mostow used families of cyclic coverings of the projective line to obtain non-arithmetic monodromy and non-arithmetic lattices. In their construction, the number of ramification points is less than the degree of the cover. In contrast, we show that if the number of ramification points exceeds the degree of the cover, the monodromy group is almost always arithmetic.
Titel Cohomology of Arithmetic Groups. Verlag Springer International Publishing.
Arithmetic Teaching Apparatus
Print ISBN In the survey paper, I wonder whether there is some way to talk about superstrong approximation for Galois groups with bounded ramification. For instance; let G be the Galois group of the maximal extension of Q which is tamely ramified everywhere, and unramified away from 2,3,5, and 7. We could ask something like the following. Given any finite quotient Q of G, and any two elements of G whose images generated Q, we get a connected Cayley graph of degree 4 on the elements of Q, by means of those two elements and their inverses.
I have no real reason to think so. The introductory talks by Frank Calegari and Nicolas are a great place to start.
I was raised to think of torsion classes in homology as a terrifying mystery that one dealt with by tensoring with the rational numbers as quickly as possible. But our knowledge about these things is actually starting to accumulate!
Here k and n are constants and N is growing. Quomodocumque Math, Madison, food, the Orioles, books, my kids.
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