Development and Applications of Non-Archimedean Analytic Geometry and Tropical Geometry

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This project concerns research in number theory to study certain properties of equations in the whole numbers. The research aims to use sophisticated modern methods in algebraic geometry and number theory to produce general bounds on the number of solutions to certain Diophantine equations. The study of Diophantine equations involves finding whole number solutions to polynomial equalities, such as when the sum of two fifth powers is again a fifth power. The study of such equations dates back almost 2,000 years and is among the most difficult problems in all of mathematics, as evidenced by the fact that it was established only 45 years ago that it is not possible to devise a general process with a finite number of operations that can decide whether a Diophantine equation has a solution. The bounds under development in this research project, which depend only on the degree of the equation (i.e., the size of the exponents), will advance knowledge in this fundamental area of mathematics. The project also involves a middle- and high-school enrichment program, in addition to support for undergraduate and graduate education. In this project, the investigator seeks to use p-adic analysis and the Chabauty-Coleman method, along with ideas from tropical and non-Archimedean geometry, to give uniform bounds (in terms of the genus) on the number of rational points on hyperbolic curves satisfying certain conditions, refining earlier results. These conditions generally involve a constraint on the Mordell-Weil rank. Using related methods, he will also attempt to prove the uniform Manin--Mumford conjecture, which gives a uniform bound (again in terms of the genus) on the size of a torsion packet on a hyperbolic curve over an algebraically closed field of characteristic zero. Ideally this result would be unconditional; as a first step, the principal investigator will treat Mumford curves and curves with compact-type reduction.