Organizer: Heng Du

Teaching language English

Description

In this mini-course, we will discuss canonical extensions of p-adic shtukas from integral models of Shimura varieties to their toroidal compactifications.

Session 1

Part 1: Shtukas for Non-Reductive Groups

In this part, we review the theory of local models and shtukas for parahoric group schemes associated with reductive groups, and explain how to generalize these results to certain smooth affine integral models of non-reductive groups. Such generalizations are essential for the study of canonical integral models of mixed Shimura varieties. We also establish several key lemmas that will be used in subsequent sessions.

Part 2: Diamonds and shtukas in the log setting I

In this part, we discuss a generalization of diamonds and shtukas to the setup of fs log schemes. We will focus on the motivation of making this kind of definitions, discuss several variants and relations with the work of other people. Some properties of log shtukas analogous to the usual shtukas will be discussed if time permits.

References:

Scholze—Weinstein: Berkeley lecture notes.

Pappas-Rapoport: p-adic shtukas and the theory of global and local Shimura varieties

Mao–Wu, Section 1-2.

Session 2

Part 1: Diamonds and shtukas in the log setting II

We continue to discuss some basic properties of the log shtukas including an equivalence of categories on the generic fiber and the rigidity of extension from the generic fiber to the integral model.

Part 2: p-adic local systems on toroidal compactifications

This is a brief introduction of toroidal compactifications of Shimura varieties. We will emphasize what the boundaries look like in general and how to compute the tower at the boundary when the level at p varies.

References:

Mao-Wu: Section 2-3.

Wu: Arithmetic compactifications of integral models of Shimura varieties of abelian type

Session 3

Part 1: Canonical Integral Models of Mixed Shimura Varieties and Their Compactifications

In this part, we review the theory of canonical integral models developed by Pappas and Rapoport and explain how to extend this framework to mixed Shimura varieties. We then introduce canonical integral models of toroidal compactifications and discuss their uniqueness and functoriality.

Part 2: Extension of shtukas to toroidal compactifications

In this part, we will introduce how to extend and glue shtukas on mixed Shimura varieties to a log shtuka on the toroidal compactification. We will also discuss how this method can be generalized in other related settings.

References:

Pappas–Rapoport, p-adic Shtukas and the Theory of Global and Local Shimura Varieties.

P. Daniels, P. van Hoften, D. Kim, and M. Zhang, On a Conjecture of Pappas and Rapoport.

Mao–Wu, Sections 4–5.

Session 4

Part 1: Construction of canonical integral models of toroidal compactifications

In this part, we discuss a construction of canonical integral models of compactifications in the abelian-type case.

Part 2: Well-Positionedness of Various Strata and Applications

In this part, we discuss applications to the geometry of the special fibers of canonical integral models. In particular, we review the theory of well-positionedness for various strata in the special fibers, following Lan–Stroh, and establish analogous results in much greater generality by studying the degeneration of shtukas along the boundary. We also present further geometric applications, including the non-emptiness of the basic locus.

References:

K.-W. Lan and B. Stroh, Compactifications of Subschemes of Integral Models of Shimura Varieties.

Mao–Wu, Section 6-7.

Prerequisites and audience

Prerequisites Familiarity with algebraic geometry, Shimura varieties, and the basics of p-adic geometry.

Target audience Graduate students and researchers in arithmetic geometry.

References

[1] Kai-Wen Lan and Benoît Stroh. Compactifications of subschemes of integral models of Shimura varieties.

[2] Peter Scholze and Jared Weinstein. Berkeley Lectures on p-adic Geometry.

[3] Georgios Pappas and Michael Rapoport. p-adic shtukas and the theory of global and local Shimura varieties.

[4] Patrick Daniels, Pol van Hoften, Dongryul Kim, and Mingjia Zhang. On a conjecture of Pappas and Rapoport.

[5] Peihang Wu. Arithmetic compactifications of integral models of Shimura varieties of abelian type.

[6] Shengkai Mao and Peihang Wu. Canonical extensions of p-adic shtukas on toroidal compactifications of Shimura varieties.