Modular Forms by Mahesh Kakde

Explore the intricacies of modular forms in this comprehensive lecture by Mahesh Kakde at the International Centre for Theoretical Sciences. Delve into key concepts including Seegel and elliptic modular forms, congruence subgroups, cusps, Fourier expansions, and fundamental domains. Examine Gauss's Theorem, Eisenstein series, and dimension formulas. Investigate holomorphic modular forms, differential operators, and theta series. Learn about Hecker's trick and explore various examples and propositions throughout the lecture. Gain insights into extremal practices and their applications in modular form theory.

Modular Forms
Definition of Seegel Modular Forms?
What are called Elliptic Modular Forms?
Remark
Congruence Subgroups
What Cusps are?
Fourier Expansion
Example - Explicitly in terms of coefficients
Remark of SL2Z modulo plus minus 1
Theorem
Fundamental Domain
Gauss's Theorem
Eisenstein series
Proposition and dimension formula
Sketch
Eisenstein series of out 2
Proposition
Proof - Hecker's trick
Holomorphic modular forms
Differential operators
Proposition
Sketch
More examples
Theta series
Proposition
Sketch
Extremal practices