Automorphic Forms and Arithmetic

I will discuss joint work with E. Lecouturier, S. Shih, and J. Wang on a construction of maps from the homology of Bianchi spaces for an imaginary quadratic field F to second K-groups of ray class fields of F. These maps are “Eisenstein” in the sense that they factor through the quotient by the action of an Eisenstein ideal way from the level. These long-expected maps are direct analogues of known explicit maps in the setting of modular curves and cyclotomic fields. We employ a substantial refinement of a method Venkatesh and I developed for constructing Eisenstein cocycles. That is, the maps are constructed as pullbacks at torsion points of certain 1-cocycles valued in second K-groups of function fields of products of two CM elliptic curves.

In joint work with A. Krause and B. Morin we give a new construction of the Bloch-Kato exponential map (or rather its inverse) using prismatic cohomology. We apply this construction to prove Conjecture C_{EP} of Fontaine and Perrin-Riou, also known as Kato’s local epsilon conjecture, for Tate motives over local fields of characteristic zero.