The Hasse-Minkowski Theorem states that a quadratic form over a number field is isotropic (i.e. it represents zero non-trivially) if and only if it is isotropic over all completions of the field. Over most fields a similar statement would not hold. However, in recent years similar local-global principles for isotropy of quadratic forms were discovered, in particular over function fields of certain two-dimensional schemes.
In several lecture series the course aims to develop a thorough understanding of the arithmetic of some special types of fields, of local-global principles for quadratic forms over these fields and for the techniques involved in proving them. The necessary background on field arithmetic, geometry and quadratic form theory will be covered. There will further be some exercise sessions and some special talks on related topics.
| Course |
|---|
| Course I: Quadratic forms over fields (Karim Johannes Becher) |
| Course II: Local-global principles in number theory and geometry (Asher Auel) |
| Course III: Field patching and local-global principles (Raman Parimala) |
| Course IV: Splitting ramification and residue characteristic two (Suresh Venapally) |