Arithmetic of Rationally Simply Connected Varieties

Polynomial equations are ubiquitous in science and engineering. Particularly important are the kinds of polynomial equations that arise in fields like cryptography and computer science, where the inputs and the outputs of the polynomials are fractions or their near-cousins (elements in a "global field"). For these equations, it is important not only to know that there are solutions in fractions, but also to know that there are many solutions, and that we can find such solutions efficiently in a finite amount of runtime (i.e., there are efficient bounds on the size of the numerators and denominators of some fraction solution). Investigating these questions is a major goal of algebraic geometry. Remarkably, ideas from geometry, particularly ideas suggested by physics, give a proof of existence of integral solutions for many special polynomial equations over global function fields. The main goal of this project is to exploit this advance and prove the veracity of certain conjectured bounds on rational solutions of these special polynomial equations ("rationally simply connected" systems of equations) over global function fields, thus giving an efficient algorithm for finding rational solutions. In addition, there are several educational and training goals: improving the writing of math students through weekly meetings, helping with a summer math camp for high school students, and holding twice-annual one-day training workshops for math students coinciding with one an important weekend workshop series in algebraic geometry. Technically, the main objective is to study the Batyrev-Manin conjecture on asymptotics of rational points of bounded height over global function fields for the special class of "rationally simply connected" varieties: a class that includes, for instance, the projective homogeneous varieties so ubiquitous in representation theory. Recent work relates rational points over global function fields to geometric properties of moduli spaces of rational curves on lifts of the varieties over the complex numbers. By exploiting this, there is hope to give efficient height bounds on rational points implied by the Batyrev-Manin Conjecture, and hopefully to settle the conjecture in important special cases. Secondary goals are to understand the Picard groups of these moduli spaces (roughly, the different possible "height functions"), and to extend the amazingly successful story of Neron models to a more general class of varieties than Abelian varieties. Broader impacts include continuing and extending a mathematical writing and professional development seminar begun by the PI, to continue the partial support of the PI for the Mathematics Summer Camp at Stony Brook University, and to institute a new series of one-day training workshops for graduate students timed to coincide with the AGNES series of twice-annual weekend workshops in algebraic geometry.