Quantitativity in the Mordell Conjecture

The Mordell conjecture asserts that there are only finitely many rational points on a smooth projective curve of genus at least two over a number field.

The uniform Mordell problem asks for suitable upper bounds on the number of rational points in the Mordell conjecture, and has been solved by combining works of Vojta, Dimitrov--Habegger--Gao and Kuhne.

In this survey, we will introduce a quantitative version of the uniformity problem proved by the recent work of Yu--Yuan--Zhou.