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TCC (Spring 2021): p-adic modular forms

This course is administered by the Taught Course Centre (TCC) and available to the Universities of Bath, Bristol, Imperial, Oxford and Warwick. Please refer to the current syllabi for registration instructions and the list of running courses.


[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.)


Instructor: Pak-Hin Lee
Email: Pak-Hin.Lee "at"
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.


  • Familiarity with modular forms, e.g. Chapters 1-5 of Diamond--Shurman's A First Course in Modular Forms
  • Familiarity with p-adic numbers, e.g. Chapter II of Serre's A Course in Arithmetic
  • Exposure to algebraic geometry in the language of schemes, and willingness to pick things up on the go


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 "" with "".)

Other supplementary references include:


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.