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.

Since the celebrated work of Wiles, Taylor--Wiles on the connection between modular forms and elliptic curves through Galois representations, studying the link between these objects has been a central subject in number theory. In such a correspondence, the weight of modular forms is matched with the Hodge--Tate weight of Galois representation. Similarly, Serre weight conjectures explain how weights of mod p automorphic forms are related to Galois representations. In this talk, I will introduce Serre weight conjectures, and we discuss a proof of the conjecture for $\mathrm{GSp}_4$ under technical assumptions using a novel geometric result on Galois deformation rings. If time permits, we also explain an application to modularity lifting for $\mathrm{GSp}_4$. This is based on a joint work with Daniel Le and Bao Viet Le Hung.

I will discuss recent results for algebraicity of critical values of Spin L-functions for $\mathrm{GSp}_6$. I will also discuss ongoing work toward the construction of p-adic L-functions interpolating these values. I will explain how this work fits into the context of earlier developments, while also indicating where new technical challenges arise. This is joint work with Giovanni Rosso and Shrenik Shah. All who are curious about this topic are welcome at this talk, even without prior experience with Spin L-functions.

The Bernstein center of a p-adic reductive group G is a beautiful and explicit commutative ring which acts on "everything" related to the representation theory of G. In recent years, the idea has emerged that this ring contains a canonical subring - the stable Bernstein center - which should be intimately related with the local Langlands correspondence. However, while it's easy to define the stable Bernstein center, it is very difficult to exhibit elements in this subring. On the other hand, recent work of Fargues-Scholze defines another totally canonical subring of the Bernstein center, whose construction uses V. Lafforgue's theory of excursion operators adapted to the Fargues-Fontaine curve. After reviewing these stories, I'll sketch a proof that the Fargues-Scholze subring is actually contained in the stable Bernstein center, for all G.