At the November workshop, I described a new algorithm to cover compact, congruence locally symmetric spaces by balls. I’ll discuss how to compute the nerve of such a covering and Hecke actions on its cohomology. Joint work with Aurel Page.
Tag - Arithmetic groups
In 1880, Markoff studied a cubic Diophantine equation in three variables now known as the Markoff equation, and observed that its integral solutions satisfy a form of non-linear descent. Generalizing this, we consider families of log Calabi-Yau varieties arising as moduli spaces for local systems on topological surfaces, and prove a structure theorem for their integral points using mapping class group dynamics. The result is reminiscent of the finiteness of class numbers for linear arithmetic group actions on homogeneous varieties, and this Diophantine perspective guides us to obtain new extensions of classical results on hyperbolic surfaces along the way.
Venkatesh has recently proposed a fascinating conjecture relating motivic cohomology with automorphic forms and the cohomology of arithmetic groups. I'll describe this conjecture, and discuss its connections with the local geometry of eigenvarieties and nonabelian analogues of the Leopoldt conjecture. This is joint work with Jack Thorne.
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