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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: 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:

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".)

Other supplementary references include:

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.