For a connected linear algebraic group defined over field there has been considerable progress in recent years in the study of homogeneous spaces over function fields of curves over a p-adic field. It has been proved recently by Parimala and Suresh that every quadratic form in at least nine variables over the function field of a p-adic curve has a nontrivial zero. Further, patching techniques developed by Harbater-Hartmann-Krashen have provided tools to study certain local-global principle for existence of rational points on homogeneous spaces over such fields under the assumption that the algebraic group is rational. We propose to study homogeneous spaces under linear algebraic groups over function fields of curves over local and global fields. The study over function fields over global fields would have tremendous consequences and could lead to the fact that every quadratic form in large enough number of variables over function fields of curves over totally imaginary number fields has a nontrivial zero. We propose to study questions of Hasse principle for existence of rational points on homogeneous spaces under connected linear algebraic groups over such fields. In the context of simply connected groups, the Rost invariant for principal homogeneous spaces is a powerful tool to study these spaces. We propose to study obstruction to the injectivity of the Rost invariant for function fields of curves over local and global fields.
The study of homogeneous spaces under linear algebraic groups includes study of several interesting algebraic objects like quadratic forms, central simple algebras, Octonian and Albert algebras. The study over number fields is enriched by class field theory techniques. A classical theorem of Hasse-Maass-Schiling for number fields is the following Hasse principle: an element in the number field is a reduced norm from a central simple algebra if and only if it is positive at all real completions where the division algebra is ramified. A similar criterion for reduced norms for 2-dimensional fields is a far-reaching extension of this classical result for number fields. We propose to replace class field theory by purely homological properties of these fields and the geometry of the associated arithmetic surfaces to study properties of homogeneous spaces in the general setting of function fields of curves over local and global fields.
Significant results: The main goal of the project is the study of algebraic groups and homogeneous spaces under connected linear algebraic groups over function fields, more specifically, over function fields of p-adic curves. The following are some of the main results proved during the project. • Degree three Galois cohomology of function fields of surfaces (jointly with V. Suresh, arxiv: 1012.5367 ). A local-global principle in degree three Galois cohomology of function fields of surfaces with respect to symbols in the Brauer group is established. This led to the following interesting consequences: 1) Every element in the degree three mod-l Galois cohomology of function fields of curves over local fields with residue field characteristic not equal to l or global fields of positive characteristic p not equal to l is a symbol. 2) The Brauer-Manin obstruction is the only obstruction to Hasse principle for the existence of zero-cycles of degree one on certain classes of surfaces over global fields of positive characteristic. • Colliot-Th'el`ene conjecture and finiteness of the u-invariants (Jointly with Max Lieblich and V. Suresh, Math Annalen, DOI: 10:1007). A major open question is the period-index bound for function fields of curves over number fields. This question is related to bounding the u-invariant of such fields. It is open whether quadratic forms in large enough number of variables over function fields of curves over totally imaginary number fields represent zero nontrivially. A conditional answer to this question has been given, modulo a conjecture of Colliot-Th'el`ene concerning the Brauer-Manin obstruction being the only obstruction to the Hasse principle for the existence of zero cycles of degree one on smooth projective varieties over number fields. This leads to challenging questions concerning the existence of zero cycles of degree one on certain twisted modulii spaces. • Higher reciprocity laws and rational points (jointly with J.-L. Colliot-Th'el`ene and V. suresh, arxiv : 1302.2377). Let K be a complete discrete valued field and F a function field in one variable over K. We introduce new obstructions to Hasse Principle for homogeneous spaces under connected linear algebraic groups over F, using the Bloch-Ogus complex. These higher ‘Brauer-Manin’ ob- struction leads to examples to show that the rationality assumption for local-global principles in the patching setting of Harbater-Hartmann- Krashen cannot be dispensed with. • Period-index questions and the u-invariant of function fields over complete discrete valued fields (jointly with V. Suresh, Invent Math DOI: 10.1007). Let F be a function field in one variable over a complete discrete valued field with residue field κ of characteristic p > 0. We prove that the index of the p-torsion division algebras over F is bounded in terms of the p- rank of κ. A corresponding result when the residue field characteristic is coprime to the period is due to Harbater-Hartmann-Krashen. Our bound is precise when the residue field is perfect, leading to the theorem that the u-invariant of F is 8, a result which goes back to Heath- Brown/Leep for p-adic curves. We also prove that the u-invariant of F is finite if the p-rank of κ is finite. Broader Impact: • Training of graduate students: The PI organised jointly with V. Suresh, a number of seminars and lecture courses to train the graduate students in the general area of number theory and algebraic geometry and more specifically on topics related to linear algebraic groups and homogeneous spaces. • Organising conferences: The PI organised a confference on ‘Ramifications in algebra and Geometry’ at Emory University during May 2011, together with Asher Auel, Eric Brussel, Danny Krashen and Skip Garibaldi. A large number of graduate students and young researchers presented their work during this conference and a very interesting problem session was part of the organisation. • Invited colloquia, Conference talks, Summer schools: The PI has been disseminating the results of the project via publications as well as delivering invited talks at conferences and workshops, at colloquia and seminars. She delivered the Coxeter lecture series at the Fields Institute, Toronto (May 2012), Noether lecture at the joint AMS-MAA national meeting at San Diego (January 2013) and the Bernoulli lecture at EPFL, Lausanne (December 2012). She also taught a minicourse during the Lens summer school on ‘Galois cohomology, Motives and cubic Jordan algebras (May 2012). This school was aimed at graduate students and young researchers.
