CAREER: Arithmetic of Surfaces

William Marsh Rice University, Houston TX

Investigators

Abstract

The PI will study the existence and distribution of rational points on algebraic surfaces defined over global fields. The proposed research has two mayor component projects. The first focuses on K3 surfaces: building on earlier work of the PI and his collaborators, and with a view towards arithmetic applications, the PI will pursue systematic, conceptual and practical methods to explicitly construct unramified Azumaya algebras representing transcendental Brauer classes on K3 surfaces. The second project focuses on del Pezzo surfaces: proving new cases of a conjecture of Colliot-Thélène and Sansuc that Brauer-Manin obstructions suffice to explain failures on local-to-global phenomena for del Pezzo surfaces, efficient computation of these obstructions on low-degree surfaces, and statistics on failures of the Hasse principle. An overarching theme in arithmetic geometry is the study of systems of polynomial equations in many variables, with the constraint that the coordinates of the solutions be rational numbers or integers (for example, 11/17, or -9). When no such solutions exist, one tries to understand the phenomena behind the absence of solutions. The geometry associated to a system of polynomials bears on the possible obstructions to the existence of solutions, and this project seeks to make such an intuition precise in some cases when a system of polynomial equations defines a surface. Although this project studies fundamental questions from a theoretical point of view, the structure of solutions to certain kinds of polynomial equations has well-documented applications (for example, in establishing secure protocols for the transmission of information between two parties that have never met). In addition to research activities, the PI will run a two-week program each summer for the duration of the grant. The goal of the program is to foster enrollment and persistence rates in STEM majors at the college level. The target demographic consists of rising 8th and 9th graders from the Houston Independent School District, including students from underrepresented groups in STEM fields. The content of the program is foundational material in the study of rational solutions to polynomial equations.