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Home » DDC - Valuation Theory: Defining valuation and holomorphy rings in function fields using quadratic forms

Seminar

DDC - Valuation Theory: Defining valuation and holomorphy rings in function fields using quadratic forms September 16, 2020 (10:00 AM PDT - 11:00 AM PDT)
Parent Program:
Location: MSRI: Online/Virtual
Speaker(s) Nicolas Daans (Universiteit Antwerpen)
Description

Valuation theory plays a major part in the interaction between number theory and logic. In this seminar, a variety of topics from valuation theory, and in particular connections to model theory, will be discussed. There will be a strong emphasis on results involving the definability of valuations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Video

Defining Valuation And Holomorphy Rings In Function Fields Using Quadratic Forms

Abstract/Media

Valuation theory plays a major part in the interaction between number theory and logic. In this seminar, a variety of topics from valuation theory, and in particular connections to model theory, will be discussed. There will be a strong emphasis on results involving the definability of valuations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

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.



Many of the ideas presented in this talk will be part of upcoming joint work with Philip Dittmann.

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Defining Valuation And Holomorphy Rings In Function Fields Using Quadratic Forms

H.264 Video 25215_28729_8502_Defining_Valuation_and_Holomorphy_Rings_in_Function_Fields_Using_Quadratic_Forms.mp4