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Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.)

We are very far from proving the existence of such an integer, let alone find an explicit value which works.

My question is:

What is the best known lower bound for $n$?

One way to obtain a lower bound $m$ for $n$ is to prove the existence of a curve of genus $2$ over $\mathbf{Q}$ with at least $m$ rational points.

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I thought the Bombieri-Lang conjecture implied the statement that the number of rational points lying on a genus $2$ curves over $\bf Q$ is bounded by a number depending on the rank of the Mordell-Weil group of the jacobian of the curve (by the work of Caporaso-Harris-Mazur). How would you prove that there is an absolute constant ? Even the conjectural "effective Mordell" does not seem to imply that. –  Damian Rössler Jul 27 '12 at 21:26
    
Theorem 1.1. in "UNIFORMITY OF RATIONAL POINTS" by CAPORASO HARRIS and MAZUR states that, assuming the weak Lang conjecture, there exists a real number $c(K,g)$ such that for all curves $X$ over $K$ of genus $g\geq 2$, the number of $K$-rational points of $X$ is bounded by $c(K,g)$. So conjecturally, there really is a uniform bound on the number of rational points. –  Harry Jul 29 '12 at 8:35
    
@Harry. Thank you for the reference. I wasn't aware of that result. –  Damian Rössler Jul 29 '12 at 10:07
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"One way to obtain a lower bound m for n is to prove the existence of a curve of genus 2 over Q with at least m rational points." Is there another way? –  JSE Aug 27 '12 at 3:16
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