Seminars & Colloquia Calendar

Abstract:  Given a variety over a field F and a collection of overfields of F, one may ask whether the existence of rational points over each of the overfields (local) implies the existence of a rational point over F (global). Such local-global pinciples are a main tool for understanding the existence of rational points on varieties. 

In this talk, we study varieties that are defined over semi-global fields, i.e., function fields of curves over a complete discretely valued field. A semi-global field admits several natural collections of overfields which are geometrically motivated, and one may ask for local-global principles with respect to each such collection. We exhibit certain cases in which local-global principles for rational points hold. We also show that local-global principles for zero-cycles of degree one hold provided that local-global principles hold for the existence of rational points over extensions of the function field. This last assertion is analogous to a known result for varieties over number fields. 

(Joint work with J.-L. Colliot-Thélène, D. Harbater, D. Krashen, R. Parimala, and V. Suresh)