Some Algorithmic Questions Related to the Mordell Conjecture

Number theory is a fundamental branch of mathematics that deals with properties of the integers. Since ancient times, people have been interested in integer solutions to equations, called Diophantine equations; the Pythagorean equation, the Pell equation, and Fermat's Last Theorem are well-known problems of this type. In modern times, research has focused on general techniques for this sort of problem.

A breakthrough result, proven by Faltings, shows that certain types of equations have only finitely many integer solutions. However, there is no known algorithm to find these solutions in general; algorithmic approaches to Diophantine equations are an active area of research. Surprisingly, this problem relates to many other areas in math, including algebraic geometry, complex analysis, Galois theory, and more.

In order to compute solutions to Diophantine problems, we need to develop methods to perform some calculations in these other fields. This project involves studying several of these difficult computational problems.

The goal of this project is to investigate various computational and algorithmic topics related to finiteness theorems in number theory. For example, a theorem of Faltings guarantees a polynomial equation of a certain type can have only finitely many rational solutions. (In geometric language, a curve of genus at least two can have only finitely many rational points.) All known proofs of Faltings's theorem are ineffective: one knows that the solutions are finite in number, but the proof provides no way to tell whether one has found them all.

Recent work, including a new proof of Faltings's theorem by the PI and Venkatesh, has opened up promising new avenues toward an algorithmic solution. The PI will study algorithmic and computational questions in algebraic geometry and number theory, motivated by Faltings's theorem and other problems in algebraic and arithmetic geometry. The PI hopes to study algorithmic approaches to the Shafarevich conjecture (determining abelian varieties over a fixed number field, with good reduction away from a fixed finite set of bad primes) and the Riemann-Hilbert correspondence (finding algebraic differential equations with given monodromy representations).

The Riemann-Hilbert correspondence, in turn, should be applicable to the problem of finding branched covers of curves, as well as the problem of studying variations of Hodge structure over a given base. Both these problems relate to Faltings's theorem.

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.