Shimura Variety Seminar
Since my original motivation for studying Shimura varieties was to investigate the relationship between motives and automorphic representations, and we are now approaching the end of the basic material in Milne’s notes, I plan to continue studying the following topics.
- Andi’s notes on Elliptic Curves and Modular Forms: The simplest example of a Shimura variety, and an opportunity to review the properties of elliptic curves
- Andi’s notes on Shimura Varieties: Part III, concerning some results over finite fields
- Kevin Lan’s notes: Example-based material that is accessible and easy to read, offering an opportunity to learn more about compactification theory
- E. E. Eischen’s notes on Automorphic Representations of Unitary Groups: To understand the basic motivations behind the research
- Milne’s Mixed Shimura: I would like to study automorphic vector bundles here
Since I am interested in Deligne’s conjecture, I also need to study some motives, but I have already forgotten étale cohomology. Ultimately, I also need to learn how to extract arithmetic and automorphic information from Shimura varieties and establish the corresponding relationships. It is known that finding motives associated with automorphic representations is intrinsically difficult.
Arthur Packet Seminar
The goal is to prove the equivalence of Arthur packets and ABV packets based on ABV theory. Endoscopic lifting is used in the proof for classical groups. After this seminar ends, we may discuss Eisenstein series.
- Adams Chapter 4: Proof of Thm 3.2.2. I need to study some of the results used here
There are also some related topics that I find thought-provoking:
- The D-module theory and microlocal theory used in ABV, as well as the classical relationship between perverse sheaves and D-modules, need to be reviewed
- How are the algorithms in Atlas implemented? Study the relevant algorithms
- Endoscopy theory: On the global side, I am particularly interested in how endoscopic classification is achieved, but this requires a thorough mastery of the trace formula
Other Thoughts
- I would like to stay at the Institute for Advanced Study over the summer and sit in on some Lean 4 courses
- Read books and write reflections on them, watch some classic films, and so on