Rational Points on Curves via Vojta's Inequality
Date/Time: Tuesdays, 10.30-12.00, during Trinity Term 2021
Room: Online on MS Teams (if possible, we might switch to in-person lectures at some point during the term) - link available upon request
Contents: I will give an overview of the history of Faltings' theorem (Mordell's conjecture), Vojta's proof of it, and subsequent applications of Vojta's method such as:
Faltings' big theorem on the distribution of rational points on subvarieties of abelian varieties of arbitrary dimension,
upper bounds for the number of rational points on a curve of genus > 1 defined over a number field,
and a height inequality of Rémond that was crucial in Habegger and Pila's proof of the Zilber-Pink conjecture for a curve in an abelian variety over the algebraic numbers.
The focus of this lecture will be a somewhat technical height inequality of Rémond, generalizing an inequality of Vojta. After a broad introduction to the topic, I will start by recalling in detail basic facts on curves, abelian varieties, and heights. I will then sketch (without giving all details) how these tools can be used to prove the Vojta-Rémond inequality and how the above results can then be deduced from this inequality together with the basic tools.
No prior knowledge is assumed apart from some basic algebraic geometry (varieties, divisors, and line bundles) and some basic algebraic number theory.
Material:
Lecture notes (last updated: 09/06/24)
Handwritten notes from the lectures (with apologies for the handwriting; post-lecture corrections in red): 27/04, 04/05, 11/05, 18/05, 25/05, 01/06, 08/06, 15/06
Notes on Faltings' proof of Mordell's conjecture "as seen by Lawrence-Venkatesh" (last updated: 25/04/21)