TCC (Spring 2021): p-adic modular forms
This course is administered by the Taught Course CentreLink opens in a new window (TCC) and available to the Universities of Bath, Bristol, Imperial, Oxford and Warwick. Please refer to the current syllabiLink opens in a new window for registration instructions and the list of running courses.
Announcements
[18 March] There will be office hours today, starting at 2 pm.
[9 March] This week's office hours have been rescheduled to 5-6 pm on 12 March (Friday).
[25 28 February] Based on the preferences received thus far, I will set up office hours on the next three Tuesdays:
- Dates: 2, 9, 16 March (Tuesdays)
- Time:
TBA12 pm to 1 pm
[22 January] If you would like to receive updates about this course, please email me with:
- Your name, institution, year
- Your backgrounds in modular forms and algebraic geometry, as well as research interests
- Whether you are taking the course for credit (Note: Due to data protection compliance I do not have access to the list of registered students.)
Logistics
Instructor: Pak-Hin Lee
Email: Pak-Hin.Lee "at" warwick.ac.uk
Time: Thursdays 2 pm to 4 pm, starting on 21 January 2021
Location: Online via Microsoft Teams
Course Description
One of the most powerful tools in modern number theory is p-adic deformations of arithmetic objects; for example, p-adic families of modular forms play a crucial role in the proofs of Fermat's last theorem and Iwasawa main conjecture. This course will cover the basic theory of p-adic modular forms as originated by Serre and Katz in the 1970's.
Prerequisites:
- Familiarity with modular forms, e.g. Chapters 1-5 of Diamond--Shurman's A First Course in Modular FormsLink opens in a new window
- Familiarity with p-adic numbers, e.g. Chapter II of Serre's A Course in ArithmeticLink opens in a new window
- Exposure to algebraic geometry in the language of schemes, and willingness to pick things up on the go
References
The main references are the articles of Swinnerton-Dyer, Serre and Katz from the Antwerp proceedings, which are available via Springer Link. (Note to Warwick students: You can access Springer Link by replacing "link.springer.com" with "0-link-springer-com.pugwash.lib.warwick.ac.uk".)
- [SD] H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, Modular Functions of One Variable IIILink opens in a new window, Springer, 1973.
- [S1] Jean-Pierre Serre, Formes modulaires et fonctions zêta p-adiques, Modular Functions of One Variable IIILink opens in a new window, Springer, 1973.
- [K] Nicholas M. Katz, p-adic properties of modular schemes and modular forms, Modular Functions of One Variable IIILink opens in a new window, Springer, 1973.
Other supplementary references include:
- [S2] Jean-Pierre Serre, Congruences et formes modulairesLink opens in a new window, Séminaire Bourbaki, Springer, 1973.
- [C] Frank Calegari, Congruences between modular formsLink opens in a new window, Notes for Arizona Winter School 2013 (video lecturesLink opens in a new window).
- [L] David Loeffler, Modular curvesLink opens in a new window, Notes for TCC course, 2014.
Schedule
The following is subject to change as the course moves along.
Slides | Date | Topics | References |
Lecture 1 | 21 Jan (Thu) | Introduction; mod p modular forms | [SD, §3] |
Lecture 2 | 28 Jan (Thu) | Mod p modular forms; higher congruences between modular forms | [SD, §3], [S1, théorème 1] |
Lecture 3 | 4 Feb (Thu) | Higher congruences between modular forms; p-adic modular forms à la Serre | [S1, §1] |
Lecture 4 | 11 Feb (Thu) |
Properties of p-adic modular forms; application to p-adic zeta functions (Note: P.21-24 contain a corrected discussion of the special values of the p-adic zeta function.) |
[S1, §1] |
Lecture 5 | 18 Feb (Thu) |
Hecke operators and application to congruences; Weierstrass parametrization (Note: The generalized lemma on P.23 isn't quite right; see Problem Sheet 2.) |
[S1, §2] |
Lecture 6 | 25 Feb (Thu) |
Geometric modular forms (Note: P.24 contains an extra discussion about Tate uniformization.) |
[K, App. 1 & §1] |
Lecture 7 | 4 Mar (Thu) | Geometric modular forms; p-adic modular forms à la Katz | [K, §1 & §2] |
Lecture 8 | 11 Mar (Thu) | p-adic modular forms; Hecke operators; canonical subgroups and applications | [K, §3], [C, §3] |
Supplementary Lecture 1 | 12 Mar (Fri), 5 pm | Geometric interpretations of p-adic modular forms | |
Supplementary Lecture 2 | 18 Mar (Thu), 2 pm | p-adic families of modular forms |
Problem Sheets
Assessment will be based on three problem sheets, due at the beginning of Weeks 6, 8 and 10 tentatively.
- Problem Sheet 1 : due on 22 February (Monday)
- Problem Sheet 2 : due on 8 March (Monday)
- Problem Sheet 3 : due on 9 April (Friday)