BIRS-PIMS TeamUp: Small Solutions to Thue Equations Over Quadratic Imaginary Fields (24rit503)

The Banff International Research Station will host the PIMS-BIRS TeamUp: "Small Solutions to Thue Equations Over Quadratic Imaginary Fields" workshop in Banff from November 24 - December 7, 2024.

One of the oldest problems in mathematics is that of finding integer solutions to polynomial equations with integer coefficients; these equations are known as Diophantine equations. These problems date back at least as far as 1800 BCE, when Babylonians compiled integer solutions to the equation a2 + b2 = c2 on the Plimpton 322 tablet. It turns out that the equation a2 +b2 = c2 has infinitely many distinct integer solutions (one of which has a = 3, b = 4, and c = 5), though this fact is not obvious.

However, not every Diophantine equation has infinitely many solutions, and subtle changes in the equations can lead to drastic changes in the behavior of solutions. For example, the equation a2 − 2b2 = 1 has infinitely many solutions, but the equation a3 − 2b3 = 1 has only two: (a, b) = (1, 0), and (a, b) = (−1,−1). Our research group studies Diophantine equations like the latter: equations that have two variables, and where the exponents on the variables are always larger than 2. We aim to show that certain classes of these equations have a very limited set of solutions.

The types of equations we study-known as Thue equations-have few direct applications, but they have some connections to cryptography via some geometric objects known as elliptic curves. For example, Garcia-Selfa and Tornero use facts about the solutions to Thue equations to solve problems about elliptic curves in their paper “Thue equations and torsion groups of elliptic curves.”