Class Diary

• Wednesday, January 21.
  • The first day of class. Handed out syllabus, talked about a number of applications of modularity.
• Friday, Jaunary 23.
  • No class.
• Monday, January 26.
  • Action of the modular group, generators, the fundamental domain, and the slash operator.
• Wednesday, January 28.
  • Class cancelled.
• Friday, January 30.
  • Eisenstein series and their Fourier expansions.
• Monday, February 2.
  • Class cancelled.
• Wednesday, February 4.
  • The valence formula.
• 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
  • Hecke operators
• Monday, March 2
  • Decomposition into character subspaces, computation of Hecke operators.
• Wednesday, March 4
  • Commutativity properties, definition of T_n 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
  • Twisting modular forms.
• 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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