A universal definition of Z in Q
Date:
Abstract: It is a long-standing open problem whether the ring of integers $\mathbb{Z}$ has an existential first-order definition in $\mathbb{Q}$, the field of rational numbers. A few years ago, Jochen Koenigsmann proved that $\mathbb{Z}$ has a universal first-order definition in $\mathbb{Q}$, building on earlier work by Bjorn Poonen. This result was later generalised to number fields and to global function fields, using classical machinery from number theory and class field theory related to the behaviour of quaternion algebras over global and local fields.
In this talk, I will sketch a variation on the techniques used to obtain the aforementioned results. It allows for a relatively short and uniform treatment of global fields of all characteristics that is less dependent on class field theory. Instead, a central role is played by Hilbert’s Reciprocity Law for quaternion algebras. Finally, I will touch on quantitative aspects of the method, and, if time allows, discuss other instances in which similar techniques can be applied.
This talk was given at the Oberseminar zur Algebra und Zahlentheorie in Augsburg. You may find the slides for this talk here.
A variation of this talk was also given at the Algebra and Number Theory Seminar at Penn State on October 7, 2020. You may find the slides for this talk here.