Automorphic Forms and Arithmetic

We will present a new notion of isogeny between ‘p-divisible groups with additional structure’ that employs the cohomological stacks of Drinfeld and Bhatt-Lurie—-in particular the theory of apertures developed in prior work with Gardner—-and combines it with some invariant theoretic tools familiar to the geometric Langlands and representation theory community, namely the Vinberg monoid and the wonderful compactification. This gives a uniform construction of p-Hecke correspondeces and Rapoport-Zink spaces associated with unramified local Shimura data. In particular, we give the first general construction of RZ spaces associated with exceptional groups. This work is joint with Si Ying Lee.

In the quantum theory of angular momentum, the Racah--Wigner coefficient, often known as the 6-j symbol, is a numerical invariant assigned to a tetrahedron with half-integer edge-lengths. The 6 edge-lengths may be viewed as representations of SU(2) satisfying certain multiplicity-one conditions. One important property of the 6j symbol is its hidden symmetry outside the tetrahedral ones, originally discovered by Regge. In this talk, we explore a generalized construction, dubbed tetrahedral symbol, in the context of rank-1 semisimple groups over local fields, and explain how the extra symmetries may be explained by relative Langlands duality. Joint work with Akshay Venkatesh.

In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we have seen many breakthroughs on constructing such systems for other Galois representations, including settings such as twisted and cubic triple product, symmetric cube, and Rankin--Selberg, with applications to the Bloch--Kato conjecture and to Iwasawa theory. For this talk, I'll consider Galois representations attached to automorphic forms on a totally definite unitary group U(2r) over a CM field which are distinguished by the subgroup U(r) x U(r). I'll discuss a new "first explicit reciprocity law" in this setting and its application to the corresponding Bloch--Kato conjecture, focusing on new obstacles which arise from the lack of local multiplicity one.