Lecturer: Prof. Eyal Goren

Time: Monday 10:30 - 11:30 and 12:30 - 13:30; Wednesday 10:30 - 11:30.

Location: Burnside Hall, Room 1214.

Exercises.

This course is meant to be an introductory course to modular forms. It is open to undergraduate students (instructor's permission required). As much as possible the exposition will be self-contained. The grade in the course will be based on assignments, reading of additional background material and a final essay.

Our entrance point to the subject is the problem of sphere packing. This problem asks for the most efficient way in which one can pack cannon-balls, say, in a ship’s hull. More abstractly we shall consider packing in arbitrary dimensions that are more structured than just any packing. This will lead us to the study of lattices.

There are beautiful and deep connections between lattices and other subjects of mathematics, which we shall investigate. In particular, we shall discuss the construction of lattices from codes and the construction of lattices from root systems. One invariant that shall be introduced is the theta function of a lattice, which encodes how many vectors of any given length the lattice has. Such a theta function is an example of a modular form.

We shall then turn to a systematic development of modular forms and modular curves. The key constructions will be done in complete detail: this includes the construction of quotients of the upper half

space as Riemann surfaces and their compactifications, the Riemann-Roch theorem and dimension formulas for modular forms, Eisenstein series and cusp forms, Hecke operators and

eigenforms. New and old forms and the connection to Galois representations (the last topic as a survey only). We shall then study complex elliptic curves in detail: Weierstrass equations and the Weierstrass uniformization, the group law on elliptic curves, the modular curves as moduli spaces for elliptic curves. Eichler-Shimura theory. The Shimura-Taniyama conjecture, Frey curves and Fermat’s last theorem (sketch only). We shall also study some spaces of modular forms in detail (for example for SL(2, Z)) and apply the results to some interesting lattices such as the E8 and the Leech lattice and to classical problems in number theory such as representations of an integer as a sum of squares.

If time allows we shall discuss Hodge structures and the Hodge conjecture and in that context re-visit the Shimura-Taniyama conjecture and discuss connections between compact Shimura curves and modular curves.

The course is to be followed by the course Modular Forms II, given by Prof. Henri Darmon in the Winter term. I can be contacted at goren at math dot mcgill dot ca for further information.