 ## MATH40260: Modular Forms of One Variable     Semester 1, 2014/15

### Dr Masha Vlasenko

G12 Science North, School of Mathematical Sciences
Belfield Office Park

Classes: Tuesday & Wednesday 10-12am

Classical (or "elliptic") modular forms are functions in the complex upper half-plane which transform in a certain way under the action of a discrete subgroup of SL_2(R) such as SL_2(Z). There are two cardinal points about them which explain why modular forms are interesting. First of all, the space of modular forms of a given weight on a given group is finite dimensional and algorithmically computable, so that it is a mechanical procedure to prove any given identity among modular forms. Secondly, modular forms occur naturally in connection with problems arising in many areas of mathematics, from pure number theory and combinatorics to differential equations, geometry and physics.

### I. Definitions and first examples

1. Introduction: a few examples from arithmetic and combinatorics
2. Definition of modular forms and construction of a fundamental domain for the modular group
3. First examples: Eisenstein series and their Fourier coefficients
4. Home assignment 1
5. Finiteness of dimensions of the spaces of modular forms
6. The discriminant function
7. Home assignment 2
8. The modular invariant j

### II. Modular forms for congruence subgroups

1. Congruence subgroups
2. Home assignment 3
3. Fundamental domains and quotiens
4. A pari/gp script which I used to check which images of elliptic points are indetified in the fundamental domain for the principal congruence subgroup of level 2.
Home assignment 4
5. Modular forms and dimension formulas
6. Further examples: theta series
Home assignment 5

### III. Moduli of elliptic curves

1. Complex tori
2. Plane curves
Home assignment 6
3. Elliptic curves
Home assignment 7
4. Complex tori as elliptic curves
Home assignment 8
5. Moduli spaces of elliptic curves

### IV. Hecke operators, eigenforms and L-series

1. Hecke operators on lattices
Home assignment 9
2. Hecke operators on modular forms
3. Eigenforms
4. L-series
Home assignment 10

## Software

PARI/GP
Sage
Magma (look at online calculator and Handbook's section on modular arithmetic geometry)

## Literature

[Serre]
J.-P. Serre, A Course in Arithmetic, Chapter VII
[1-2-3]
The 1-2-3 of Modular Forms, Elliptic Modular Forms and Their Applications by D.Zagier
[Zagier]
From Number Theory to Physics, Introduction to Modular Forms by D.Zagier
[Milne MF]
J.S. Milne, Modular Functions and Modular Forms
[Milne EC]
J.S. Milne, Elliptic Curves
([pdf] from Milne's web page)
[DiamondShurman]
F. Diamond, J. Shurman, A First Course in Modular Forms
[Lang]
S. Lang, Introduction to Modular Forms
[PanchishkinManin]
Yu.I.Manin, A.A.Panchishkin, Introduction to Number Theory, Part II: Ideas and Theories, Chapters 6 and 7