TITLE: Uniform Faltings
QUESTION [10 upvotes]: Suppose I give you positive integers $g\geq 2$ and $N.$ Is it always possible to find an absolutely irreducible curve of genus $g$ over $\mathbb{Q}$ which has at least $N$ rational points? For that matter, what if $g=2?$ I assume that the answer is YES, but what do I know?

REPLY [16 votes]: On the contrary, some conjectures suggest that the answer is NO! It follows from the Bombieri-Lang conjecture (sometimes known as Lang's conjectures) that a uniform bound should exist. 
More precisely, given a number field $F$ and a genus $g\geq2$ BL implies that there's some bound $N(F,g)$ such that every smooth projective genus $g$ curve over $F$ has at most $N(F,g)$ points. Even better -- the bound depends only on the degree of $F$ over $\mathbb{Q}$, not $F$ itself. 
The fact that Bombieri-Lang implies this was proved by Caporaso, (Joe) Harris and Mazur in 1997 and my recollection at the time was that some people regarded this as evidence against Bombieri-Lang rather than for the uniform bound. However, at this stage, the question is open, even for curves of genus 2 over $\mathbb{Q}$. 
I once saw a talk of Elkies where he exhibited a curve of genus 2 over $\mathbb{Q}$ with something like 588 rational points, but this may not be the record any longer.