Zero-Cycles over Arithmetic Fields and Reciprocity Laws

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A central question in almost every science is the classification of objects that feature similar characteristics. In algebraic geometry, mathematicians are interested in the classification of algebraic varieties, that is, sets of solutions to polynomial equations. This classification is carried out by determining and computing certain invariants of the variety. Such invariants can be numerical, or more often, sets that have a specific algebraic or geometric structure. The more geometric approach focuses on algebraic varieties over the complex numbers, which themselves have a rich geometry. In number theory, on the other hand, mathematicians are interested in finding integer or rational solutions to polynomial equations; a famous example in this area is Fermat's last theorem. The set of rational numbers is very scarce within the set of complex numbers, which is what makes such solutions so hard to detect. This project aims to study a geometric invariant, called the Chow group of zero-cycles, that relates both to the classification problem and to the arithmetic of rational solutions. The main goal of the project is to investigate conjectures that deal with the structure of this group when we work over the rational numbers or over the arithmetic analog of the real numbers, namely the p-adic numbers. The methods in this project will involve techniques from arithmetic and algebraic geometry as well as K-theory. The project focuses on the study of abelian varieties, a class of varieties that has some extra structure. The project involves "local questions" for varieties over the p-adic numbers, where the use of p-adic Hodge theory will be the key. Among the main goals of the local program is to establish a conjecture of Colliot-Thélène. Second, the project involves also "global questions" for varieties over the rational numbers. The goal of the global program will be to prove a conjecture of Beilinson. Lastly, the project will investigate a "local-to-global" program related to obstruction questions. The goal is to prove a conjecture of Kato and Saito, which could potentially lead to the construction of a new type of Euler system.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.