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Explore advanced techniques for studying the cohomology of Shimura varieties through the lens of Igusa stacks in this mathematical lecture. Delve into the Igusa stack diagram, a powerful framework that expresses Shimura varieties with infinite level at p as fiber products of Igusa stacks with flag variety diamonds over the moduli stack of G-bundles on the Fargues-Fontaine curve. Examine how this Cartesian diagram enables the application of categorical local Langlands program techniques to analyze Shimura variety cohomology. Learn about significant applications including torsion-vanishing results, Eichler-Shimura relations, and Ihara's lemma, drawing from research by Koshikawa, Hamann-Lee, Daniels-van-Hoften-Kim-Zhang, Yang-Zhu, and Yang. Discover ongoing work on the relative intersection cohomology of Igusa stacks, representing collaborative research between the speaker and colleagues including Linus Hamann. This lecture forms part of a comprehensive mini-course series covering the construction and applications of Igusa stacks across various levels of generality, making it essential viewing for researchers working in arithmetic geometry, the Langlands program, and related areas of algebraic number theory.
