A genus 2 family with 226 sections

Abstract

Faltings' theorem [Fal83],[Fal91] (formerly the Mordell conjecture [Mo22]) states that a curve of genus greater than one over any number field has only finitely many points. Again a natural question is how many points can such a curve have. Caporaso, Harris, and Mazur [CHM97] have shown that the weak Bombieri-Lang conjecture implies that for any number field $F$ and any integer $g \ge 2$ there is an absolute upper bound $B(F; g)$ on the number of points on a genus $g$ curve over $F$. Furthermore, the strong Bombieri-Lang conjecture implies that for each genus $g \ge 2$, there is an absolute bound $C(g)$ depending on the genus -- but not on the field -- such that over any number field, only finitely many curves of genus $g$ have more than $C(g)$ points. Again we can ask what those two bounds are and, as it turns out, it helps to consider families that come from K3 surfaces. Specifically, we will consider the case $g = 2$. We use a K3 surface $X$ that is a double cover of $P^2$ ramified over a smooth sextic curve $C$, so every pencil of lines gives us a family of genus 2 curves. Any line that is tangent to $C$ at 3 points will lift to a pair of curves on $X$ that become sections of the family given by any pencil of lines. In section 3 we will construct an K3 surface (over a number field) and corresponding sextic with 64 such tritangents. Furthermore, there will turn out to be other rational curves in $P^2$ of higher degree that also meet $C$ only at tangent points. By suitably choosing the pencil and performing suitable base changes we find a family of genus 2 curves with 226 sections. This is the current best record, the previous [Elk06] being 150 sections, which remains the record for a family over $\Q$.