Description:

About a decade ago Scholze discovered that the Siegel modular varieties (i.e. moduli spaces of polarized abelian varieties with level structure) become perfectoid when passing to infinite level, and that there is a Hodge-Tate period map from the resulting perfectoid space to a flag variety. He used the geometry of the period map to prove important p-adic properties of torsion cohomology classes of locally symmetric spaces, and deduced the existence of Galois representations attached to such classes; this was a major breakthrough in the Langlands program. Later, Caraiani and Scholze used their refinement of the Hodge-Tate period map to prove important vanishing properties of torsion cohomology classes of certain Shimura varieties. The proof was simplified by Koshikawa by relating the problem to the cohomology of local Shimura varieties. Even more recently, people have got a clearer understanding of the relationship between the Hodge-Tate period map and the geometrization of local Langlands via the Fargues-Fontaine curve, in particular the groundbreaking work of Fargues-Scholze. From this point of view, more general vanishing results for torsion cohomology classes are being proved.

In this course we will start with the proof of Scholze’s theorem that the Siegel modular varieties become perfectoid at infinite level. We will then discuss the important geometric properties of the Hodge-Tate period map as in the works mentioned above. When we move to more recent developments and more advanced topics, we will still try to give as many proofs as possible.

Prerequisite:

Understanding of fundamental concepts in algebraic geometry and algebraic number theory is needed. Basic understanding of modular curves and abelian varieties will be assumed. Prior experience with p-adic Hodge theory will be extremely helpful. Some acquaintance with adic spaces and perfectoid spaces is strongly preferred but not strictly necessary, as we will review them.

Biography:

Yihang Zhu is a Professor at Yau Mathematical Sciences Center. He obtained his Ph.D. from Harvard University in 2017. His research centers around the Langlands program, especially Shimura varieties and trace formula methods.