what is the maximum number of rational points of a curve of genus 2 over the rationals - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T19:16:03Z http://mathoverflow.net/feeds/question/103327 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103327/what-is-the-maximum-number-of-rational-points-of-a-curve-of-genus-2-over-the-rati what is the maximum number of rational points of a curve of genus 2 over the rationals Dirk 2012-07-27T16:32:14Z 2012-07-27T17:08:39Z <p>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.)</p> <p>We are very far from proving the existence of such an integer, let alone find an explicit value which works.</p> <p>My question is:</p> <p>What is the best known lower bound for $n$?</p> <p>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. </p> http://mathoverflow.net/questions/103327/what-is-the-maximum-number-of-rational-points-of-a-curve-of-genus-2-over-the-rati/103331#103331 Answer by Kestutis Cesnavicius for what is the maximum number of rational points of a curve of genus 2 over the rationals Kestutis Cesnavicius 2012-07-27T17:08:39Z 2012-07-27T17:08:39Z <p>I believe it is 642. See <a href="http://www.mathe2.uni-bayreuth.de/stoll/recordcurve.html" rel="nofollow">http://www.mathe2.uni-bayreuth.de/stoll/recordcurve.html</a></p>