Defining valuation and holomorphy rings in function fields using quadratic forms
Date:
Abstract: Quadratic forms have played a role in some recent definability results in number theory. For example, Jochen Koenigsmann’s celebrated construction of a universal first-order definition of $\mathbb{Z}$ in $\mathbb{Q}$ - building on earlier work by Bjorn Poonen - relies on a number of classical facts from number theory on the behaviour of quadratic forms over $\mathbb{Q}$, in particular the local-global principle for isotropy due to Minkowski.
One can make abstraction of the required results on quadratic forms over $\mathbb{Q}$ and aspire to find results analogous to Koenigsmann’s in other classes of fields where the quadratic form theory is well-understood, e.g. through a local-global principle. As a teaser for this idea, this talk will give a sketch of how Kato’s local-global principle for 3-fold quadratic Pfister forms for function fields in one variable over local or global fields can be used to find existential or universal definitions of holomorphy and valuations rings of such function fields.
This talk was given at the online workshop Decidability, definability and computability in number theory at SLMath (formerly MSRI) in Berkeley, California. You may find the slides for this talk here and the recording here.
Please take note: it was pointed out to me that parts of this talk misrepresent (by omission) the state of the art of research at the time, in particular on Hilbert’s 10th Problem over function fields. Furthermore, the presented research was expanded on and can be found in a manuscript co-authored with Karim Johannes Becher and Philip Dittmann titled “Uniform existential definitions of valuations in function fields in one variable”. arXiv link