Modular Forms I
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.
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
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.