Let P be an irreducible polynomial over the rational numbers of degree 5. Faltings proved in 1983 that the number of solutions in Q of the equation y^2=P(x) is finite, confirming thus Mordell's conjecture (from 1923). (Faltings was awarded the Fields medal in 1986 and the Abel prize in 2026.)

Is that number bounded independently of P ? This open problem is the uniform version of Mordell's conjecture.

In a more appropriate setting, Faltings' theorem asserts that the number of points (over a fixed number field $K$) of an algebraic curve of genus >1 is finite. An unproved uniform version would bound that number in terms of $g$ and $K$ only. (The notions of genus and and number field will not be necessary to follow the lecture, which will be non-technical.)

We will describe some progresses and exceptions on this question and move to the analogous questions on elliptic curves (given by y^2=P(x), with P polynomial of degree 3) where more is known.