Back to Departmental Colloquium: Fall 2021
Departmental Colloquium
Date: Thursday, Oct 7, 2021
Time: 4:00PM
Location: JWB 335
Jennifer Balakrishnan
Boston University
Title |
Rational points on curves and quadratic Chabauty |
Abstract |
Let C be a smooth projective curve defined over the rational numbers with genus at least 2. It was conjectured by Mordell in 1922 and proved by Faltings in 1983 that C has finitely many rational points. However, Faltings’ proof does not give an algorithm for finding these points, and in practice, given a curve, provably finding its set of rational points can be quite difficult. In the case when the Mordell–Weil rank of the Jacobian of C is less than the genus, the Chabauty–Coleman method can be used to find rational points, using the construction of certain p-adic line integrals. Nevertheless, the situation in higher rank is still rather mysterious. I will describe the quadratic Chabauty method (developed in joint work with N. Dogra, S. Müller, J. Tuitman, and J. Vonk), which can apply when the rank is equal to the genus. I will also highlight some examples of interest, from the time of Diophantus to the present day. |
Computational Mathematics
Applied Mathematics