Welcome to MathOverflow

Question

7

1

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.

flag	asked Jul 27 at 16:32 Dirk 71●1

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 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 at 8:35

@Harry. Thank you for the reference. I wasn't aware of that result. – Damian Rössler Jul 29 at 10:07

1

"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 at 3:16

Kestutis Cesnavicius · Answer 1 · 2012-07-27 17:08:39Z UTC

10

I believe it is 642. See http://www.mathe2.uni-bayreuth.de/stoll/recordcurve.html

answered Jul 27 at 17:08

Kestutis Cesnavicius
1,198●5●12

what is the maximum number of rational points of a curve of genus 2 over the rationals

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.