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: TBA 12 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)