Abstracts for Number theory lunch seminar

Noncommutative tori can be viewed as limits of elliptic curves for which the period lattice degenerates to a pseudo-lattice (a rank-2 free subgroup of the real line). A noncommutative torus whose period pseudo-lattice correspond to an order in a real quadratic field is called a real multiplication noncommutative torus. Based on strong analogies with the case of elliptic curves with complex multiplication Y.

The quotient of a curve by a correspondence usually reduces to a point in algebraic geometry. One way to fix this pathology is to extend algebraic geometry in a "non-commutative" direction. Another way (which is the subject of this talk) is to extend algebraic geometry by staying within the commutative setting but adjoining instead a new operation: the Fermat quotient. It turns out that in this new geometry a number of interesting quotients of curves by correspondences become non-trivial and indeed rather interesting.

We give a new local characterization of the Local Langlands Correspondence, using deformation spaces of $p$-divisible groups, and show its existence by a comparison with the cohomology of some Shimura varieties. This reproves results of Harris-Taylor on the compatibility of local and global correspondences, but completely avoids the use of Igusa varieties and instead relies on the classical method of counting points a la Langlands and Kottwitz.

The Rost invariant is an invariant of degree 3 of linear algebraic groups. Its existence was conjectured by Serre and proved by Rost in the beginning of 90s. In his paper "Cohomologie galoisienne" Serre showed that the coprime components of the Rost invariant of any group of type F4 over a purely transcendental extension of degree 1 of a p-adic field satisfy a non-trivial relation. Then he asked about other relations existing between coprime components. In the talk I will give another example of such relation.