Rigid meromorphic cocycles for orthogonal groups

Virtual talk.
In the early 2000s, Darmon initiated a fruitful study of analogies between Hilbert modular surfaces and quotients Y := SL_2(ZZ[1/p]) \ H x H_p, where H is the complex upper half plane and H_p is Drinfeld's p-adic upper half plane.  As Y mixes complex and p-adic topologies, making direct sense of Y as an analytic space seems difficult.  Nonetheless, Y supports a large collection of exotic special points - corresponding to the units of real quadratic fields which are inert at p - and Darmon-Vonk have described an incarnation of meromorphic functions on Y, so called rigid meromorphic cocycles.

This talk describes joint work with Henri Darmon and Lennart Gehrmann, in which we study generalizations Y' of the space Y to orthogonal groups G for quadratic spaces over QQ of arbitrary signature QQ.  The spaces Y' support large collections of exotic special points - corresponding to subtori of G of maximal real rank - and we define explicit rigid meromorphic cocycles on for Y'; these RMCs are analogous to meromorphic functions on orthogonal Shimura varieties with prescribed special divisors first studied by Borcherds, and they generalize the RMCs constructed by Darmon-Vonk.  We will also discuss some computations suggesting that values of our RMCs at special points might realize new instances of explicit class field theory.