Part I. Singular moduli for real quadratic fields. (Joint with Alice Pozzi and Jan Vonk.)

The theory of complex multiplication asserts that the values of classical modular functions like the $j$-function at imaginary quadratic arguments of the Poincaré upper half-plane generate abelian extensions of  imaginary quadratic  fields, and admit explicit factorisations. There is a rich literature devoted to proving 
similar results for the  CM values of  holomorphic  and nearly holomorphic modular forms (Shimura), higher Green’s functions (Duke-Li, Ehlen, Viazovska),  certain mock modular forms (Bruinier, Ono, …), etc. 
I will discuss an analogous theory   in which the  modular objects are   so-called {\em rigid meromorphic cocycles} on a Drinfeld upper half plane. Their main virtue is that they can be meaningfully evaluated at real quadratic irrationalities, leading to  an ``explicit class field theory"  for real quadratic fields.