CAREER: New Directions in p-adic Heights and Rational Points on Curves

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Determining whole number solutions to polynomial equations has been an active area of study for at least two millennia. Nevertheless, many questions remain, and these equations continue to be crucially important, as the techniques used to study them have helped shape the foundation of modern cryptosystems. In 1922, Louis Mordell conjectured that equations defining curves of genus at least 2 have only finitely many rational solutions. Gerd Faltings proved this in 1983, but his proof does not explicitly yield the set of rational points on these curves. Algorithmically determining this set is one of the most fundamental open problems in number theory. Quadratic Chabauty is a new approach to determining the set of rational points, and through a combination of theoretical and computational strategies, the PI will give quadratic Chabauty algorithms to determine rational points on new classes of curves. This project also includes several educational and outreach components, including a collection of undergraduate-focused workshops in Guam on the topic of computational tools, aimed at broadening participation of traditionally underrepresented groups in STEM. The PI will also co-organize a week-long summer program in mathematical exploration and computation for high school students in the Boston area, as well as a semester program at Mathematical Sciences Research Institute on Diophantine geometry.The main research themes are centered on algorithms for determining rational points on curves of genus at least 2, using p-adic heights.They include the following: using p-adic heights to produce a quadratic Chabauty algorithm for modular curves, developing an elliptic quadratic Chabauty algorithm to study twisted Fermat curves, and building infrastructure in Coleman integration and p-adic heights in families. These new algorithms will be run on large databases of curves, and the resulting data will be analyzed and shared with the mathematical community. This has the potential to yield new insight into refined hypotheses under which theorems can be proved, as well as more precise conjectures to investigate.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.