糖心TV

This course is administered by the (TCC) and available to the Universities of Bath, Bristol, Imperial, Oxford and 糖心TV. Please refer to the 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
• Familiarity with p-adic numbers, e.g. Chapter II of Serre's
• 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 糖心TV 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, , Springer, 1973.
• [S1] Jean-Pierre Serre, Formes modulaires et fonctions z锚ta p-adiques, , Springer, 1973.
• [K] Nicholas M. Katz, p-adic properties of modular schemes and modular forms, , Springer, 1973.

Other supplementary references include:

• [S2] Jean-Pierre Serre, , S茅minaire Bourbaki, Springer, 1973.
• [C] Frank Calegari, , Notes for Arizona Winter School 2013 ().
• [L] David Loeffler, Modular curves, 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, 搂3]
Lecture 2	28 Jan (Thu)	Mod p modular forms; higher congruences between modular forms	[SD, 搂3], [S1, th茅or猫me 1]
Lecture 3	4 Feb (Thu)	Higher congruences between modular forms; p-adic modular forms 脿 la Serre	[S1, 搂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, 搂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, 搂2]
Lecture 6	25 Feb (Thu)	Geometric modular forms (Note: P.24 contains an extra discussion about Tate uniformization.)	[K, App. 1 & 搂1]
Lecture 7	4 Mar (Thu)	Geometric modular forms; p-adic modular forms 脿 la Katz	[K, 搂1 & 搂2]
Lecture 8	11 Mar (Thu)	p-adic modular forms; Hecke operators; canonical subgroups and applications	[K, 搂3], [C, 搂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)