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.

Hida theory provides a p-adic interpolation of modular forms that have a property known as ordinary. The corresponding 2-dimensional p-adic Galois representations are known to be reducible when restricted to a decomposition group at p. Which ordinary modular forms satisfy the stricter property that the restriction to the decomposition group at p is both reducible and decomposable? We propose a length 1 "bi-ordinary" complex, built out of overconvergent modular forms of critical slope, whose cohomology we show supplies a satisfactory answer to this question. We also use a degree-shifting action on this cohomology to deduce a p-adic realization of a Stark unit group on some parts of weight 1 coherent cohomology. This is joint work with Francesc Castella.

The determination of the unitary dual of a Lie group is a longstanding problem. In this talk I will explain how the unitarity of a representation of a real reductive group can be read off from its Hodge filtration establishing a conjecture made by Wilfried Schmid and myself a while back. This is part of a larger set of conjectures which postulate that global sections of mixed Hodge modules on flag manifolds give rise to mixed Hodge structures. In joint work with Dougal Davis we have proved a substantial part of these conjectures and in particular the Hodge theoretic unitarity criterion. Hodge theory also allows us to treat other notions of representation theory, such as lowest K-types, geometrically. Along the way we prove general results about mixed Hodge modules which are of interest on their own.