Duality Theorems in Arithmetic Geometry and Applications
Bilinear Structures in Theory and Practice
Duality theorems are at the heart of class field theory both
for number fields and geometric objects like curves and abelian
varieties. They relate abelian Galois extensions with invariants
of the base object. In particular, class groups of rings of
integers and group schemes attached to Jacobians of curves
are involved in this game. Since these groups are the most
popular for producing crypto primitives based on discrete
logarithms (which use a priori only the Z-linear structure)
they carry unavoidably a bilinear structure. In the first
lecture we want to sketch the mathematical background and
destructive and constructive consequences.
In the second lecture we describe the Lichtenbaum-Tate pairing and
explain why Brauer groups of local fields are closely related to
torsion points of (generalized) Jacobian varieties over finite fields.
This includes theoretical as well as computational aspects
From Curves to Brauer Groups
Computing in Brauer Groups of Number Fields
Finally we use local classifield theory to describe the Brauer
group of local fields by cyclic algebras classified by their invariant.
We explain how this is related to the classical discrete logarithm
in finite fields. By the celebrated Hasse-Brauer-Noether-Sequence
we can globalize and find an Index-Calculus-Algorithm to determine
local invariants. We shall apply this both to the discrete logarithm
in finite fields and to the computation of the Euler totient function.
Thematic Program Home page
The Fields Institute Coxeter Lecture Series (CLS) brings a leading
mathematician to the Institute to give a series of three lectures
in the field of the current thematic program. The first talk is
an overview for a general mathematical audience, postdoctoral
fellows and graduate students. The other two talks are chosen,
in collaboration with the organizers of the thematic program,
to target specialists in the field.