MATH 600 Modular forms (Topics in Algebra)

Final presentations:

• Modular forms of half-integral weight Notes by K. Buzzard
• Elliptic integrals; connection with differential equations. Abel-Jacobi map. Resources:
  • Varadarajan's notes on elliptic functions ;
  • Notes on Abel-Jacobi map
• Algebraic theory of modular forms; possibly the statement of Eichler-Shimura correspondence
(see Diamond and Shurman and Chapter II in Milne's notes).
• Any topic that we do not cover in class from Zagier's notes in "The 1-2-3 of modular forms: lectures at a summer school in Nordfjordeid, Norway" (available online through UBC library).
• Any computational topic (not covered in class) from William Stein's book.
• The monster group and moonshine
• Explicit class-field theory; j-invariant; approximations of pi. A paper and another one about computing the digits of pi using modular forms.

Detailed Course outline

• Tuesday September 4 : NO CLASS -- Imagine day and qualifying exams. Graduate Orientation at 4:30pm.
• Thursday September 6 : Lecture 1. Overview and motivation; the definition of modular forms; Riemann surfaces; elliptic functions (approximately matching Milne's Introduction).


• Tuesday Sep. 11 : Lecture 2. The goal (to be achieved next class) is to define the structure of a Riemann surface on "H mod Gamma". We did: 1. A quick survey of actions of topological groups (see Milne, Section 1 (pp. 13-14) and Part 1 of the homework 1) with the end result that "H mod Gamma" has Hausdorff quotient topology;
  2. Classification of Fractional-linear transformations, with an aside on realizing the Riemann sphere as the projective line over C (see homework). (Milne, pp.30-31, not including Remark 2.11 which will be discussed later).
• Thursday Sep. 13 : Lecture 3: 1. Realizing H as a quotient of SL_2(R). (Milne, Proposition 2.1 pp.25-26) 2. the fundamental domain for Gamma(1). (Milne, pp.32-33).


• Tuesday Sep. 18 : Lecture 4: The complex structure on H mod Gamma(1). Cusps and H^* mod Gamma(1). (Milne, pp.35-37, up to but not including the genus computation).
• Thursday Sep. 20 : Lecture 5: The complex structure on H^* mod Gamma(N). Review of the preliminaries on complex analysis on Riemann surfaces. Ringed spaces; sheaves, differential forms. (Milne, pp.16-18).
• Friday Sep. 21 (extra class) : Lecture 6: Review of the analysis/geometry, continued: Riemann-Roch theorem and the notion of genus (Milne, pp.18-23).
  See an illuminating mathoverflow discussion of Riemann-Roch,
  as well as Terry Tao's blog .


• Tuesday Sep. 25 : Lecture 7: Riemann-Hurwitz formula; the genus of the modular surface X(N). (Milne, pp.37-39).
• Thursday Sep. 27 : NO CLASS.


• Tuesday October 2: NO CLASS.
• Thursday October 4 : STILL NO CLASS!


• Tuesday October 9 : Lecture 8. Weierstrass p-function and the structure of an elliptic curve on C/\Lambda. Definition of Eisenstein series. (Milne, pp.43-47).
• Thursday October 11 : Lecture 9. lattices, revisited. Equivalence of categories: (lattices mod scaling); (compact Riemann surfaces of genus 1); (elliptic curves over C). Started discussion of modular functions and modular forms; proved Eisenstein series are modular forms and the j-invariant is a modular function. Approximately pp.41-43 and 47-50 in Milne.
  If unfamailiar with discriminants, check out this concise and useful note (or any classic algebra textbook).


• Tuesday October 16: Lecture 10. Modular forms as differentials on H/Gamma. Dimension of the space of modular forms of weight k. Zeroes of modular forms. (Milne, pp.50-54). Also, pay special attention to Proposition 4.3 and Remark 4.4 -- we finished the proof of all the statements in Remark 4.4.
• Thursday October 18: : Lecture 11. Eisenstein series G_2 and G_3 generate the ring of all modular forms. Fourier expansion of Eisenstein series. (Approximately pp. 54-57 of Milne). For Fourier expansions of Eisenstein series, see also: Notes by V.S. Varadarajan pp.1-5 for a discussion of Euler's identities.


• Tuesday October 23 : Lecture 12. The Fourier expansions of the discriminant and the j-invariant. The Jacobi's product formula for the discriminant; Dedekind's eta-function and the Eisenstein series E_1. (See a short section on pp.57-58 of Milne, and note that we covered Sections 2.3 (skipping the proof of Proposition 6) and 2.4 of Zagier's notes, not including Proposition 8.) An overview of Ramanujan's tau-function and Ramanujan conjecture (will get back to it in some detail later in the course, OK if you missed it now). An overview of some remarkable properties of the j-invariant. There is no specific reference for this part of the lecture but see Milne's Introduction section (p. 11) and the wikipedia article related to the approximations of pi using the j-invariant. See also the links above (in the "suggested presentation topics") for more in-depth treatment of the mentioned topics. This part will not be used later in the course.
• Thursday October 25 : Lecture 13. The metric and measure on the upper halpf-plane. Estimates for cusp forms and their coefficients. Petersson inner product. References: Milne, pp. 62-64, or Lang's book on modular forms, chapter 1 and section 2.1; for estimates on cusp forms, see also Zagier's notes, Proposition 8 on p.23.


• Friday October 26 : Lecture 14. Hecke operators for the full modular group. Reference: Milne, pp. 67-79 (we will eventually cover all that's mentioned in these pages, but maybe not quite in the same order). See also Lang's "modular forms" book, chapter 1.


• Tuesday October 30: : Lecture 15. 1. The proof that for the full modular group, the eigenvalues of Hecke operators are algebraic integers (Milne, pp.78-79, Lemma 5.25 to Prop. 5.27, inclusive). 2. Modular forms for congruence subgroups: recap of the dimension formulas for M_k(Gamma) and S_k(Gamma); reminder about cusps; beginnig of the construction of Eisenstein series for Gamma(N). The main reference for this part is Diamond-Shurman, section 4.2 See also Milne, pp.64-66).
• Thursday November 1: Finished Eisenstein series for Gamma(N); started Hecke operators for Gamma_1(N). Diamond-Shurman, sections 4.3 and 5.1, 5.2


• November 6-8: finish defining the Hecke operators; summarize the matter of oldforms, newforms, and Atkin-Lehner theory (some subset of sections 5.2 -5.8 of Diamond-Shurman).


• November 13-15: NO CLASS (I'll be at the Sixth Abel Conference .)
  Please look the problem set on Dirichlet series . All the solutions (and more) are contained in Titchmarsh's classic book (see Chapter 9; the whole chapter is highly recommended reading).


• November 20: Finished the discussion of Newforms; L-function attached to a modular form, and functional equation. Reference: Diamond-Shurman, sections 5.8, 5.9, 5.10. I also made some comments on Forier transform, summarized here ; please take a look.
• November 22: Many statements and no proofs: the modularity theorem; l-adic representations; connection with automorphic forms.


• November 27-29: Presentations:
  • Tuesday Nov. 27: (Ashvni?) Hecke correspondence. (the algebraic-geometric definition of Hecke operators).
  • Thursday Nov. 30: Federico, Eichler-Shimura correspondence.
• Two days of presentations in December (schedule TBD -- please fill out the poll ).