BC Number Theory & Representation Theory Seminar, Fall 2026

The endoscopic Fundamental Lemma is a formula computing certain orbital integrals for a group G in terms of orbital integrals for related endoscopic groups. A remarkable amount of geometry has arisen from this equation, culminating in Ngo Bao Chau's proof of the Fundamental Lemma almost 30 years after it was conjectured. Spencer Leslie more recently proposed endoscopic fundamental lemmas associated to symmetric varieties X=G/H instead of to a reductive group G. In this talk, I will discuss the geometric spaces and techniques that go into a proof of the case of a Siegel form of the unitary Friedberg-Jacquet variety. In particular, I will introduce Higgs bundles and affine Springer fibers for this symmetric variety, and will discuss how the local and global pictures interact. This is based on joint work with Spencer Leslie.

Let $f$ be a cuspidal Hilbert modular eigenform of parallel weight 2, without CM. The Selmer groups for the adjoint $p$-adic Galois representations attached to $f$ have a rich history, including (to name just a few) works of Flach, Wiles, Taylor-Wiles, Kisin, and more recently Newton and Thorne. In particular, these Selmer groups are now all known to be finite. However, it is still not known in general that their lengths are given by appropriate $L$-values or congruence numbers, as predicted by the Tamagawa number conjecture. In this talk, I will report on joint work in progress with Chris Skinner, in which we bound the adjoint Selmer group for $f$ in terms of the congruence number in new cases. Most notably, we are able to get a near-sharp bound even when the level of $f$ is arbitrarily bad at $p$. The proof uses a system of auxiliary Galois cohomology classes, constructed by combining an idea of Skinner and Venkatesh with level-raising techniques inspired by bipartite Euler systems.

In this talk I will describe some new and old results regarding the distribution of rational points on the unit n dimensional sphere. I will discuss effective counting problems on counting rational points with bounded denominators that fall in a fixed, or shrinking target set. These problems can be studied using tools from homogenous dynamics, modular forms, and moments of incomplete Eisenstein series.