Number theory II.a - Modular forms

MATH-511

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In this course we will introduce core concepts of the theory of modular forms and consider several applications of this theory to combinatorics, harmonic analysis, and geometric optimization.

During the course we will learn:

  • Basic definitions and facts of the theory of modular forms
  • Combinatorial properties of the Fourier expansions of modular forms
  • Applications of modular forms to harmonic analysis
  • Modular forms and the sphere packing problem

Assessment methods
70% of the final grade are awarded for the final exam and 30% of the grade come from the homework done during the semester.


Important announcements
The exercise sessions start at 8:15 and the lectures start at 10:15.

20 February

Modular beasts and where they live

  • Modular forms are everywhere
  • First examples of modular forms
  • Geometry of the upper half-plane

Reading
[1] 1.1 First definitions and examples
[2]  1. Basic definitions
[3] 1.4 The Fourier expansion of Eisenstein series. 1.5 The modular group



27 February

Eta, theta, and partitions

  • Pentagonal numbers theorem
  • Jacobi triple product expansion
  • Modularity of eta and theta functions

Reading
M. Hall, Combinatorial theory,  Chapter 4. Partitions. Section 4.1

6 March

Eta, theta, and partitions

  • Pentagonal numbers theorem
  • Jacobi triple product expansion
  • Modularity of eta and theta functions

Reading
M. Hall, Combinatorial theory,  Chapter 4. Partitions. Section 4.1



13 March

Ping-pong for the principal congruence subgroup of level 2

Principal congruence subgroup of level 2 Modular lambda function
Reading:
[1] Sections 2.1, 2.3, 2.4

20 March

Modular curves
  • Modular curves as Riemann surfaces
  • Elliptic points
  • Cusps

Reading:
[1] Section 2


27 March

Genus of a modular curve
  • Ramification and branching points
  • Degree of a holomorphic map
  • Topological genus
  • Riemann-Hurwitz formula
  • Genus of a modular curve

Reading:
[1] Section 3.1


27 March

Dimension formulas

  • Meromorphic differentials
  • Holomorphic differentials
  • Riemann-Roch formula


Reading:
[1] Section 3.1, 3.2, 3.3


10 April


Dimension formulas

  • Riemann-Roch formula
  • Dimensions of spaces of modular forms of even weight


Reading:
[1] Sections 3.4, 3.5


17 April


Complex tori, Elliptic curves, and Modularity

  • Complex tori as Riemann surfaces
  • Weierstrass  \wp -function
  • Complex tori as Elliptic curves
  • Moduli space of complex tori
  • Modular forms as homogeneous lattice functions
To read:
[3] 1.2 Elliptic functions
[1] Chapter 1: Modular Forms, Elliptic Curves, and Modular Curves.

24 April

Easter break

1 May

Petersson inner product and Hecke operators


  • Petersson inner product
  • Poincare series
  • Hecke operators
  • Hecke L-function

To read:

[1]  Sections 5.1, 5.2

[2]  Section 4. Hecke Eigenforms and L-series

[3] Chapter 6. Hecke operators


8 May

Lattices

  • Basics: lattices, the dual lattice, Gram matrix
  • Integral and even lattices
  • level of even lattice
  • theta function


Reading:
[2] Section 3.2


15 May

22 May

Extremal lattices
  • Siegel's theorem
  • Extremal lattices as sphere packings
  • Mallows, Odlyzko, Sloane "There are only finitely many extremal lattices"

To read:
[2] Section 3.2
Course on lattices by Noam Elkies. Lecture on November 4 and 6

22 May

Modularity of Theta functions of integral weight

  • Decomposition  M_k(\Gamma_1(N)) = \oplus_{\chi \mod{ N}}{M_k(\Gamma_0(N), \chi)}
  • Proof of Hecke-Schoeneberg Theorem

5 June

Shifted theta series and their transformation law

  • Shifted theta series
  • Weil representation
  • Theta kernel
  • Gauss sums

To read:
Elliptische Funktionen und Modulformen, Max Koecher, Aloys Krieg, Springer, Berlin 2007,
Course on lattices by Noam Elkies. (Lecture on December 2)

8 May

This Thursday is a public holiday. There will be no lecture and no exercise session this week.

I wish you all a nice holiday and see you next week.

Weil representation


  • Heisenberg group
  • Schrodinger representation
  • Action of SL2
  • Weil representation


Zoom sessions on Thursday 03.06:

  • 8:15 Exercise session
  • 10:15 Lecture