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Faltings' theorem [Fal83],[Fal91] (formerly the Mordell conjecture [Mo22])states that a curve of genus greater than one over any number field has onlyfinitely many points. Again a natural question is how many points can such acurve have. Caporaso, Harris, and Mazur [CHM97] have shown that the weakBombieri-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 ona genus $g$ curve over $F$. Furthermore, the strong Bombieri-Lang conjectureimplies 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 numberfield, 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 toconsider families that come from K3 surfaces. Specifically, we will considerthe 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 afamily of genus 2 curves. Any line that is tangent to $C$ at 3 points will liftto a pair of curves on $X$ that become sections of the family given by anypencil of lines. In section 3 we will construct an K3 surface (over a numberfield) and corresponding sextic with 64 such tritangents. Furthermore, therewill turn out to be other rational curves in $P^2$ of higher degree that alsomeet $C$ only at tangent points. By suitably choosing the pencil and performingsuitable 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, whichremains the record for a family over $\Q$.

A genus 2 family with 226 sections