Class Diary
- Wednesday, January 21.
- The first day of class. Handed out syllabus, talked about a number of
applications of modularity.
- Friday, Jaunary 23.
- Monday, January 26.
- Action of the modular group, generators, the fundamental domain,
and the slash operator.
- Wednesday, January 28.
- Friday, January 30.
- Eisenstein series and their Fourier expansions.
- Monday, February 2.
- Wednesday, February 4.
- Friday, February 6.
- Dimensions formula, and properties of the j-function.
- Monday, February 9.
- Transformation law for E2.
- Wednesday, February 11.
- Transformation law for the eta function, bounds for the Fourier
coefficients.
- Friday, February 13
- Congruence subgroups and indices.
- Monday, February 16
- Action of congruence subgroups on cusps, modular curves.
- Wednesday, February 18
- Extended example relating Eisenstein series, eta quotient,
and generating functions for representation as sum of four squares.
- Friday, February 20
- The V(d)-operator, Sturm's theorem, and the
conclusion of the example began on Wednesday.
- Monday, February 23
- Started random aside about p | a(p).
- Wednesday, February 25
- Finished random aside, started talking about action
of double cosets.
- Friday, February 27
- Monday, March 2
- Decomposition into character subspaces, computation
of Hecke operators.
- Wednesday, March 4
- Commutativity properties,
definition of Tn for all n.
- Friday, March 6
- Multiplicativity of the coefficients of Delta,
the Petersson inner product.
- Monday, March 9
- Adjoints of Hecke operators, basis of eigenforms.
- Friday, March 20
- Oldforms and newforms, the strong multiplicity one theorem.
- Monday, March 30
- The Fricke involution, and the K operator.
- Wednesday, April 1
- Analytic continuation and functional equation for
L-functions of modular forms.
- Friday, April 3
- Monday, April 6
- Weil's converse theorem and Eisenstein series.
- Wednesday, April 8
- General definition of L-functions
and examples. Start of discussion of Rankin-Selberg
L-functions.
- Friday, April 10
- Beginning of the proof of the analytic continuation and functional
equation of Rankin-Selberg L-functions.
- Monday, April 13
- Conclusion of the discussion of Rankin-Selberg L-functions.
- Wednesday, April 15
- Non-existence of Siegel zeroes of the symmetric square L-function.
- Friday, April 17
- Symmetric power L-functions and the Sato-Tate conjecture.
- Monday, April 20
- Symmetric power L-functions for higher level forms,
and CM forms.
- Wednesday, April 22
- Introduction to the Langlands program.
- Friday, April 24
- Modular forms mod l and the non-existence of congruences for
the partition function.
- Monday, April 27
- Swinnerton-Dyer's Theorem 2, parts (i)-(iii).
- Wednesday, April 29
- Some commutative algebra and part (iv) of Swinnerton-Dyer's Theorem 2.
- Friday, May 1
- Lemmas needed for Kiming and Olsson's theorem.
- Monday, May 4
- Proof of Kiming and Olsson's theorem and the start of
Ahlgren and Boylan's theorem.
- Wednesday, May 6
- Conclusion of the proof of Ahlgren and Boylan's theorem.
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