Summer School
Summer School – Quadratic Forms in Chile 2018
the week prior to the conference: 2nd – 5th of January 2018
Tuesday 2nd of January – Friday 5th of January 2018, the week just prior to the conference.
The first day and a half will be dedicated to introducing the basics of the theory of quadratic forms. For students with a background in quadratic forms (and who cannot arrive by January 2), it should be no problem to join the school starting from the afternoon of January 3.
The course will introduce the basic notions corresponding to quadratic forms over fields and local rings, such as isotropy, the notion of residue forms of unimodular forms over a local ring, as well as lifting of isotropy of the residue form to the completion of the ring. Moreover, ground work for the other two courses will be layed, for example the relation between orderings of fields, sums of squares, quadratic forms and valuations, or the relation between the isotropy of 3 and 4-dimensional quadratic forms and the splitting of certain quaternion algebras. Also, quadratic forms over particular base fields will be discussed without complete proves (e.g. Tsen-Lang theory for complex function fields, local-global principles over number fields or p-adic function fields).
The explicit objective of this course is the revise some better and also lesser known results on sums of squares in and quadratic forms over function fields of real varieties, such as upper and lower bounds for the so called Pythagoras number; in particular for algebraic (and arithmetic) surfaces over the real numbers – (the Pythagoras number beeing the smallest natural number n needed to express any totally positive element as a sum of n squares).
These will be consequences of implicite objectives of the course: Relating arithmetic objects and techniques for function fields (such as valuations, orderings, sums of squares) to geometric objects and techniques (such as blowing ups, divisors, double covers, generic hyperplane sections, possibly vector bundles and their chern classes).
The main objective of this course is to construct counterexamples to the existence of a local-global principle for isotropy of quadratic forms over function fields of complex algebraic varieties, with emphasis on the case of curves and surfaces. The construction of quadratic forms that are locally isotropic but globally anisotropic will involve input from the Picard group, in the case of curves, and the Brauer group and Hodge theory, in the case of surfaces. These methods invoke many of the arithmetic and geometric concepts outlined in Course 2, and will be further developed in Course 3.
More information to come.
The Brauer group of varieties may detect nonrationality of vareties. It may also prevent local-global principles for rational points of projective varieties and for integral points of affine varieties. There are «higher» analogues of the Brauer group. I shall describe various techniques to compute the Brauer group and these higher variants.
If you want us to help you find accommodation in Santiago, let us know in the registration mail, also indicate if you are potentially interested in sharing an apartment (on AirBnB) with other participants, so that we can team you up and rent the place for you (in this case we would ask you to please send a copy of your flight tickets).