Quadratic forms are algebraic objects with particularly nice algebraic and geometric properties. Local-global principles for quadratic forms are a classical topic of algebra and number theory. The Hasse-Minkowski Theorem formulates such a local-global principle for the case of a number field or a function field of a curve over a finite field. It can for example be used to show that any sum of squares in a number field is a sum of four squares.
While quadratic forms are interesting to study over general fields, it is only for very special fields that isotropy can be determined by a local-global principle. Such a local-global principle is usually expressed in terms of completions (or henselisations) of a field with respect to valuations. In the last two decades, a new technique called field patching has led to the discovery of a series of new cases of fields where certain types of quadratic forms satisfy a local-global principle. This includes in particular function fields of curves over complete discretely valued fields, such as the fields of p-adic numbers.
When a local-global principle is present, it can also shed light on other features of a field. This applies in particular to the study of field invariants such as the u-invariant or the Pythagoras number. Recent breakthroughs establishing upper bounds on such invariants for particular fields have been achieved in this way. Also the study of Hilbert’s 10th Problem has seen recent progress based on certain local-global principles.
The aim of the summer school is to introduce the attendees to this active research area, with an emphasis on the applicability of local-global principles. This will include providing the context for different scenarios of application, such as the study of the u-invariant and the Pythagoras number of fields as well as of Hilbert’s 10th Problem. Also the nuances between different types of local-global principles, for example whether formulated in terms of discrete or of more general valuations, as well as the notorious case of characteristic 2, will be highlighted. Finally, examples of failure of local-global principles will be examined.
The summer school will consist of 16-18 lectures, which are accompanied by exercise sessions and some research talks. Some familiarity with basic quadratic form theory as well as with basic valuation theory will be assumed, but also refreshed during the first days.