Algebra and its Applications (Spring 2005)

The Shafarevich-Tate group of an elliptic curve is an important invariant, measures the failure of the local-to-global principle. The second part of the Birch and Swinnerton-Dyer conjecture gives a conjectural value of the order of the Shafarevich-Tate group. In recent joint work with Loic Merel, we used the theory of visibility to give some theoretical evidence towards this conjecture. We shall describe this using an example, after introducing all the necessary concepts.

This is the first of a series of talks devoted to an introduction to the basic facts on Modular Forms.

Let M be a compact 3-manifold with non empty boundary. The space of allhyperbolic structures on the interior of M (i.e. Riemannian metrics with curvature equals -1), is wellunderstood. Its boundary is a topic of much research. We show that a certain type of hyperbolic manifolds (which are easy to understand) are dense on the boundary. This is a joint work with Richard D. Canary (Ann-Arbor).

This is the second of a series of introductory talks on Modular Forms. After we are finished talking about the fundamental domain of the modular group, topics to be covered are: lattices, elliptic functions, and their relations to modular forms.

We hace seen how a lattice function of weight 2k (a function on the space of lattices in C of weight 2k) corresponds to a modular function of corresponding weight. Therefore we need a good supply of lattice functions in order to produce modular forms. These are provided by the Eisenstein series. In this part we'll show how to produce them in terms of elliptic functions and elliptic curves. In particular, we will obtain them as Laurent coefficients of Weierstraß ℘ function near z=0 in C.

We have seen the definition and regularity properties of the Eisenstein series associated to a lattice in C. In this part, carrying over from last meeting, we'll show how to produce them in terms of elliptic functions and elliptic curves. In particular, we will characterize Eisenstein series as Laurent coefficients of Weierstraß's ℘ function near z=0 in the complex plane.

If time permits, we'll start looking into theorems characterizing the ring M_k(Γ(1)) of modular forms of weight 2k—dimension, ring structure, etc.

In this talk we will compute the dimension of M_k(Γ(1)), the vector space of modular forms of weight 2k, and characterize the ring structure of the direct sum M(Γ(1)).

More about the structure of the ring M(Γ(1)) and the modular invariant j.

We will look at a few classical expansions of modular forms for the modular group Γ(1). After having covered these classical results, we will start looking at the (compactified) quotient of the upper half plane by (a subgroup of) the modular group, and introduce the appropriate Riemann surface structure.

Introduction to the modular curves X(N) for the congruence subgroups Γ(N) of the Modular group Γ(1).