Special Session: Shimura Varieties & Arithmetic
AMS Spring Sectional — Boston College, Mar 28–29, 2026

TBD

The Bernstein center of a \(p\)-adic group is a commutative ring of certain distributions on the group, and it interacts closely with the group’s representation theory. Fargues and Scholze provide an abstract construction of a class of elements of the Bernstein center called excursion operators, which encode a candidate for the (semisimplified) local Langlands correspondence. In this talk, I will present an approach to understanding excursion operators concretely as distributions on the group, with a special emphasis on the case of \(G = \mathrm{SL}_2\) where everything can be made quite explicit.

TBD

TBD

The cohomology of global Shimura varieties is of fundamental importance in number theory, as it is the only known geometric realization of the global Langlands correspondence over number fields. In recent years, the use of techniques in geometric or categorical local Langlands have become of fundamental importance in understanding its structure. In particular, the formalism of Igusa stacks introduced by Zhang and the categorical local Langlands conjecture of Fargues-Scholze/Zhu, can be used to give a formula for the cohomology in terms of the moduli stack of local \(L\)-parameters that witnesses enough about its structure to establish many important results such as vanishing, Ihara's lemma, and the Eichler-Shimura relationship.

Our understanding of these formulas can in turn be further refined by rewriting them in terms of moduli stacks of global L-parameter, as in the conjectures of Emerton-Gee-Hellman and Zhu, as well as by considering variants for different incarnations of the cohomology of the Shimura variety such as the intersection cohomology of its minimal compactification. In this talk, we will survey some of these recent developments and explain the emerging conceptual picture. This is based on joint work in progress with Caraiani and Zhang and Bertoloni Meli, Caraiani, Koshikawa, and Zhang.

TBD

The Harris-Viehmann conjecture establishes a parabolic induction formula on the cohomology groups associated to non-basic local Shimura data. It follows that all supercuspidal representations on a Shimura variety are concentrated along the basic locus, making the conjecture relevant to the Langlands program. Historically, many cases of the Harris--Viehmann conjecture have been approached with the additional condition of Hodge--Newton reducibility on the underlying local Shimura datum. Building on previous work by Mantovan (EL/PEL case) and Hong (Hodge case), we extend the proof of the conjecture to non-basic local Shimura data of abelian type under the assumption of Hodge--Newton reducibility. We leverage Shen’s construction of Rapoport--Zink spaces of abelian type. This is joint work with Xinyu Zhou.

I will discuss work on growth of Betti numbers in small degrees in towers of Shimura varieties, as the level varies. The results are proved in the framework of the endoscopic classification of representations.

Let \(r:G_F\to \mathrm{GL}_2(\overline{\mathbb{Q}}_p)\) be a global Galois representation associated to an automorphic form \(\pi\), where \(p\) is inert in \(F\). One can construct a unitary Banach space representation \(\pi(r)\) of \(\mathrm{GL}_2(F_p)\) associated to \(r\). The \(p\)-adic Langlands Program predicts that \(\pi(r)\) depends only on \(r_p\), which is \(r\) restricted to \(p\).

Let \(V\) be the algebraic representation corresponding to the Hodge--Tate weights of \(r_p\). Then, we have \(\pi_p\otimes V^\vee\hookrightarrow\pi(r)\), and the latter induces a unitary Banach space structure on the closure of the image. This unitary Banach space structure is the same as completing \(\pi_p\otimes V^\vee\) with respect to the \(\mathrm{GL}_2(F_p)\)-invariant integral lattice induced by the completed cohomology with integral coefficients. One would first show such lattice depends only on \(r_p\), but even this is currently unknown.

Inside \(\pi_p\), we have a type \(\sigma(\tau)\), the \(\mathrm{GL}_2(\mathcal{O}_K)\) representation that determines the Bersnstein components of \(\pi_p\). Breuil conjectured that the \(\mathrm{GL}_2(\mathcal{O}_K)\)-invariant integral lattice inside \(\sigma(\tau)\otimes V\) induced by the completed cohomology with integral coefficients is uniquely determined by \(r_p\). This conjecture provides evidence for the \(p\)-adic Langlands correspondence and the local-global compatibility. In this talk, we will present our recent result on Breuil's Lattice Conjecture for higher Hodge--Tate weights, which are small relative to \(p\(.

If time permits, we will give the key ideas behind the proof. One is a structure theorem for mod \(p\) representations of \(\mathrm{GL_2}(\mathcal{O}_K)\), which are residually multiplicity free and of finite length. Another is an explicit computation of universal framed Galois deformation rings, which parameterize potentially crystalline lifts with fixed tame inertial types and higher Hodge--Tate weights.

TBD

TBD

TBD

TBD

TBD

TBD

TBD

TBD