Algebra and its Applications (Spring 2007)

Ettore Aldrovandi: Aspects of Hermitian geometry of algebraic curves and Riemann surfaces

The "fiber at (arithmetic) infinity," of an arithmetic surface, that is, the set of its complex points equipped with the standard complex manifold structure, is a classical Rieman surface. In his seminal paper Le déterminant de la Cohomologie, Deligne introduces a map which assigns to two hermitian line bundles a one-dimensional Hermitian vector space. Its Hermitian form is given by a real number built from the two bundles' Green's functions associated to their Chern forms.

We want to informally discuss two points:

• Deligne's map can be computed purely algebraically, using a candidate for an arithmetic version of motivic cohomology, and
• connections with hyperbolic geometry in 2 and 3 dimensions, and the classical uniformization theorem.

Ettore Aldrovandi: Introduction to Deligne Cohomology

Continuing from the previous two talks of this series, we give an informal introduction to the formalism of cohomology of abelian sheaves, with a particular emphasis on Deligne cohomology and some of its variants. We will stress a "hands-on" approach, and highlight geometric applications.

Ettore Aldrovandi: Cup products and Hermitian Deligne Cohomology

We wrap up what we heve done with Deligne Cohomology. First, we introduce higher degree versions of the tame symbol map using cup products, and, second, we introduce the Hermitian variant of all the above. We will interpret the Deligne symbol associated to two Hermitian line bundles introduced during the first two talks as a cup-product in the Hermitian variant of the theory.

Amod Agashe: Modular forms

This will be an elementary talk meant as a preparation for the talk by Eknath Ghate next week in the algebra seminar, which will involve modular forms. We will discuss some basics about modular forms without assuming any background beyond that of the audience. If time permits, we might talk about the connection between modular forms, abelian varieties, and Galois representations.

Eknath Ghate: Deformation theory and Galois representations

Unlike the global Galois representation attached to an ordinary newform of weight at least two, the associated local representation is reducible. Greenberg has asked whether it is semi-simple.

In the CM-case, this is known to be true, but the non-CM case is much more mysterious.

Using the deformation theoretic methods of Mazur we will give the first non-trivial examples of non-CM Hida families where no weight 2 or more member has a semi-simple representation.

This is joint work with Vatsal.

Ettore Aldrovandi: Introduction to Stacks and Gerbes

Gerbes are a particular kind of stacks, enjoying properties analogous to those of principal bundles. As such, they are the geometric objects needed to do (nonabelian) cohomology in degree 2 (and beyond), as opposed to 1.

In this series we plan to give an introduction to these ideas, assuming no previous background on the matter. (Having attended last fall Noohi's lectures will help, however.) If time permits, we plan to arrive at the point of discussing how the theory of cup product in Deligne cohomology discussed in the previous talks leads to the construction of specific examples