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.

Exam paper with solutions(published after the exam on May 18)

Syllabus

  1. Introduction: modular forms arising in elementary and advanced geometry, combinatorics and physics
  2. A supply of modular forms
  3. Families of elliptic curves and modular forms
  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:

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