The seminar will meet weekly at the Student Lounge or Thesis Room, from 4:00pm-5:00pm on Friday. The official homepage for the seminar is here.
I will be talking about various aspect of classical modular forms, especially its application in number theory. My first goal is to give you a fundamental treatment of modular forms, a first course if you will (which mean that it comes with little pre-requisite and could be viewed as a typical graduate course like algebraic topology).
The (incomplete list of) topics I want to talk about are:
- Definition of modular forms; most importantly, precise definition of various notions such as cusps, Fourier expansion and holomorphic at cusps; work out examples of modular forms notes
- Interpretation of modular forms as differential on modular curves notes
- Dimensions of the vector space of modular forms notes (for a Review of Riemann surfaces)
- Hecke operators and the theory of new forms
- L-functions and converse theorems
- Algebraic theory: algebraicity of eigenvalues of Hecke eigenforms, p-adic modular forms
- Applications of modular forms: Galois representation and non-abelian class field theory
Along the way, I shall review some topic such as Riemann surfaces, sheaf and algebraic geometry.
Unlike last semester, I shall upload the combined note when I think they are ready.
This time, I make all my notes in Kindle size.
Modular Forms I
September 7th, 2018
Abstract: Definition of modular forms, examples, classification of linear fractional transformations
Modular Forms II
September 14th, 2018
Abstract: Absolute convergence of Eisenstein series (missing from last time), cusps, elliptic points, Fourier expansion
Modular Forms III
September 21st, 2018
Abstract: Review of Riemann surfaces: the category of Riemann surfaces, uniformization theorem, compact Riemann surfaces, differentials
Modular Forms IV
September 28st, 2018
Abstract: Modular curves as Riemann surfaces, interpretation of modular forms as differential on modular curves
Modular Forms V
October 5th, 2018
Abstract: Dimension formulas for the space of modular forms
Modular Forms VI
October 12th, 2018
Abstract: Dimension formulas for the space of modular forms (Continued)
Modular Forms VII
October 19th, 2018
Modular Forms VIII
October 26th, 2018
Abstract: Hecke operators
Modular Forms IX
October 26th, 2018
Abstract: Primitive forms
Modular Forms X
November 2nd, 2018
NO SEMINAR. I WILL SPEAK AT JNTD.