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

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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.

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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.

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