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.

I will explain a conjectural description of the 'etale local structure of p-adic integral models for general Shimura varieties with level structures "of type Gamma_1(p)". This aims to generalize a classical result about the integral model of the modular curve X_1(p) established by Deligne-Rapoport over 50 years ago. The description uses two key ingredients: A new construction connecting local models of Shimura varieties for parahoric levels with toric varieties, and certain Galois covers which are toric compactifications of Lang torsors. This is work in progress, joint with Rapoport.

The Weil representation plays a central role in Langlands duality, especially in the theory of the theta correspondence. I will describe a geometrization of the Weil representation, which will be a category with a projective action of (the loop group of) the symplectic group; upon taking trace of Frobenius, it recovers the usual Weil representation. This categorical representation can be defined "over the sphere spectrum" (I will explain what this means). It turns out that the cocycle which obstructs the linearization of the action of the symplectic group can be built from Bott periodicity and the "J-homomorphism" (a classical construction in algebraic topology), and this can be viewed as a homotopy-theoretic source of the anomaly-vanishing condition appearing in relative Langlands duality. (Interestingly the cocycle in question also appears in symplectic geometry, via the Maslov index.) This is part of a story about studying (geometric) Langlands duality with coefficients in ring spectra/generalized cohomology theories, and I hope to say a bit about this broader picture.