BC Number Theory & Representation Theory Seminar, Fall 2026

We give a proof of the Tate conjecture for abelian fourfolds over finite fields. This was the first resolution of the conjecture for all abelian varieties of a fixed dimension over finite fields since the work of Tate in the 1960s. The proof relies on techniques from Ancona's proof of the standard conjecture of Hodge type for abelian fourfolds, and ultimately reduces to Markman's results on the algebraicity of Weil classes on complex abelian varieties. As an application, we complete the proof of the standard conjectures for abelian fourfolds over arbitrary fields.

Let $A$ be an $m \times n$ matrix and $b$ an $m$-vector. Twisted Diophantine approximation studies approximations of pairs $(A,b)$ when $A$ is fixed. We present a new geometric idea in metric uniform twisted approximation. Using all successive minima of the diagonal lattice trajectory associated with matrix $A$, we count lattice points, accounting for clustering in every direction. Ratios of these counts give exact Hausdorff dimensions of sets of twisted $\psi$-Dirichlet vectors for a large class of functions $\psi$, including all except finitely many power functions. We obtain Hausdorff dimension formulae for twisted singular vectors and prove that taking the intersections causes no dimension drop. We also obtain Hausdorff dimensions of level sets of the uniform Diophantine exponent.

This talk is based on a joint paper with Taehyeong Kim

We study the distinction of the Steinberg representation of a split $p$-adic group $G$ with respect to a split symmetric subgroup $H$. We relate this problem to the problem of determining the existence of a non-zero harmonic function on a certain hypergraph related to $X = G/H$. We show that the Steinberg representation is distinguished only if $X$ is quasi-split and that $\mathrm{GL}_{2n+1}/(\mathrm{GL}_{n} \times \mathrm{GL}_{n+1})$ is essentially the only example where $X$ is quasi-split but the Steinberg representation is not distinguished. As a corollary we obtain a case of relative local Langlands correspondence, we show that the Steinberg representation is distinguished if and only if its parameter factors through the dual group of $X$.

Computing rational points on nonsplit Cartan modular curves is a difficult open problem. The most successful method — Quadratic Chabauty — can so far only solve this problem for prime levels $N = 13, 17, 19$. In this talk I will report on joint work in progress with Isabel Rendell, in which we complete the first step of Quadratic Chabauty, namely the computation of a basis of de Rham cohomology and of the Hodge filtration, for levels $N = 23, 29$. Crucially, we avoid using a plane model of the curve, but instead phrase everything in terms of modular forms.

The microlocal point of view says that the study of an object on a space benefits from systematic constructions involving the cotangent bundle. It was introduced by M. Sato in the 1960s for studying partial differential equations and has since spread to many other fields. In this talk, I will discuss the basics of the microlocal theory of sheaves in various contexts. Applications will be mentioned.

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.

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.

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.