Lectures in course MATH-511: Number theory II.a - Modular forms

Canonical Divisors and Modular Forms

Covers canonical divisors on Riemann surfaces and properties of modular forms.

Meromorphic Differentials and Modular Forms

Explores meromorphic differentials on Riemann surfaces and modular forms on congruence subgroups.

Modular Forms: Dimension Formula

Explores modular forms, discussing pullback maps, meromorphic differentials, and the Riemann-Roch theorem.

Modular Forms: Dimension Formulas

Covers dimension formulas for modular forms and related proofs using Riemann-Roch corollaries.

Pentagonal Number Theorem and Jacobi Identity

Covers the pentagonal number theorem, Jacobi identity, and modularity of eta and theta functions.

Extremal Lattices

Explores extremal lattices, emphasizing bounds on shortest vectors and unique modular forms.

Lattices: Properties and Theta Functions

Explores lattices in Rd, Gram matrices, theta functions, and modular properties.

Modular curves: Riemann surfaces and transition maps

Covers modular curves as compact Riemann surfaces, explaining their topology, construction of holomorphic charts, and properties.

E8 Lattice Optimality

Delves into the construction and optimality of the E8 lattice as the densest sphere packing in dimension eight.

Petersson Inner Product and Hecke Operators

Covers the Petersson inner product and Hecke operators in modular forms theory, exploring their definitions and properties.

Modular Lambda Function: Properties and Applications

Explores the modular lambda function, its properties, and applications in modular forms.

Principal Congruence Subgroup of Level 2

Explores the principal congruence subgroup of level 2 and its group structure.

Modular Curves: Genus and Mapping Theorems

Explores holomorphic maps, ramification points, and the genus of a modular curve.

Vector Valued Functions

Explores vector valued functions, their representations, uniqueness, and congruence subgroups.

Representation of SL2Z

Delves into the representation of SL2Z and the function F, exploring its transformation properties and relation to modular forms.

Theta functions: Properties and Transformations

Explores the properties and transformations of theta functions, including modular forms and lattice levels.

Heisenberg Group: Representation

Covers the Heisenberg group representation, including shifts, multiplications, and the Fourier transform, extending to finite fields and metaplectic groups.