# MA342J: Introduction to Modular Forms

## Hilary term, 2012

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.
We start with the analytic base of the theory of modular forms, prove finiteness of dimensions and construct enough examples, such as Eisenstein series, theta series and eta-products. In the second part of the course we study families of elliptic curves, view modular curves as their moduli spaces and show how modular forms naturally arise in this context. At the end we discuss application of modular forms to Fermat's last theorem.

## Syllabus

1. Introduction: modular forms arising in elementary and advanced geometry, combinatorics and physics
• Magma script counting points on the elliptic curve y^2+y=x^3-x^2 over finite fields. It shows that numbers of points over extensions of a field satisfy a simple recursion. You can copy-paste it and run in the magma online calculator. Try to change the prime, check that p=11 is exceptional. Try to change the curve.
2. A supply of modular forms
• Definitions and first examples: Eisenstein series
• Further examples: the discriminant function and cusp forms
• Computation of dimensions of the spaces of modular forms
• Modular groups and modular forms of higher levels, modular curves, dimension formulas and Riemann-Roch Theorem
• Tutorial 3, February 23
• Study week assignment. (This doesn't count with the homework. Question 1 is to do on one's own and successful submission of question 2 will be awarded by +10% to the final mark.)
• Homework 4, due on March 8
• Play with the Fundamental Domain Drawer by Helena Verrill
• More examples: theta series
3. Families of elliptic curves and modular forms
• Affine and projective plane curves
• Elliptic curves as projective cubics
• Literature: [Milne EC]
• Tutorial 4, March 15 & March 21: transforming an elliptic curve into a Weierstrass form
• Elliptic curves over complex numbers and complex tori of dimension 1. Modular curves viewed as moduli spaces.
• Families of elliptic curves
4. Discussion of Fermat's last theorem and modularity theorem for rational elliptic curves

## Assessment

Final exam questions will be chosen from home assignments. The final mark will be 70% of the exam mark + 30% of the home assignments mark or 100% of the final exam mark, whichever is higher.

## Learning Outcomes:

On successful completion of this module, students will be able to:
• give examples of modular forms, namely Eisenstein series, theta series and eta-products
• use dimension formulas to prove various identities between modular forms
• analyze families of elliptic curves with the help of modular forms

## Magma

Online calculator
Handbook section on modular arithmetic geometry

## Literature

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