On lattices, Euclid, and zeta functions

Speaker: Valeriia Starichkova 
Affiliation: University of New South Wales

Abstract

I would like to talk about a project in progress that we started this year. Let K denote a number field and R its ring of integers. We can think of K as a vector space over rationals Q with some finite dimension n (i.e. K is Q^n) and of R as a lattice over integers Z (i.e. R is Z^n).

In 1977, Lenstra constructed a criterion for R to be Euclidean. This criterion connects the discriminant of R, the “packing” information about R, and some more specific information related to the units of R. In particular, this criterion implies a bound for the discriminant of K, which, assuming the Generalised Riemann Hypothesis, holds only if the degree of K (number n) is not too large. This fact attracted our attention at the very beginning of the project, by making a link between Lenstra's criterion and the analytic number theory.

I will introduce the required theory on my way, explain Lenstra’s criterion and talk about tools from different maths areas which are all combined in this criterion; namely, we will cover some properties of number fields, a little bit of analytic number theory, and a tiny bit of packing theory.