Algebraic Geometry Seminar: "Local-to-global principles for zero-cycles"

Abstract: Let X be a smooth projective variety over an algebraic number field F. We say that X satisfies Weak Approximation if  the set X(F) of all F-rational points on X is dense inside the product of all X(F_v), where v runs through the set of all places of F. That is, in some appropriate sense, every family of local points can be "approximated" by a global point. By the foundational work of Y. Manin, the Brauer group of X often obstructs Weak Approximation and for certain classes of varieties this obstruction is known or conjectured to fully explain the failure. In other words, every family of local points that is orthogonal to the Brauer group can be approximated by a global point. In this talk I will discuss analogs of these questions for the Chow  group CH_0(X) of zero-cycles on X. In the 1980's  Colliot-Thelene, Sansuc, Kato and Saito conjectured that the Brauer group should give the only obstruction to Weak Approximation for zero-cycles when X is a general smooth projective variety. This conjecture has only been established for certain classes of rationally connected varieties, and there is some recent partial evidence for products of K3 surfaces. The purpose of this talk will be to give some evidence for a product of elliptic curves.  Part of this talk will be based on recent joint work with Toshiro Hiranouchi, and another part is work in progress with an Appendix by Angelos Koutsianas.

Host: Humberto Diaz