Curves of genus 2 with many rational points via K3 surfaces (abstract only)

Abstract

Let C be a (smooth, projective, absolutely irreducible) curve of genus g ≥ 2 over a number field K. Faltings [Fa1,Fa2] proved that the set C(K) of K-rational points of C is finite, as conjectured by Mordell. The proof can even yield an effective upper bound on the size #C(K) of this set (though not, in general, a provably complete list of points); but this bound depends on the arithmetic of C. This suggests the question of how #C(K) behaves as C varies. Following [CHM], we define for each g ≥ 2 and K: B(g, K) = max C #C(K),with C running over all curves over K of genus g;(so infinitely many C have N rational points over K, but only finitely many have more than N ); and 2 N (g) = max K N (g, K). It is not known whether either B (g, K) or N (g) is finite for any g, K; even the question of whether N (2, Q) < ∞ is very much open. Caporaso, Harris and Mazur proved [CHM] that Lang's Diophantine conjectures [La] imply the finiteness of B(g, K), N (g) for any number field K and integer g ≥ 2; but the proof yields no estimates on these bounds. While giving upper bounds seems hopeless at present, lower bounds are more tractable: we need only construct curves or families of curves with many rational points. We announce several new constructions, all using K3 surfaces of maximal Picard number. Specifically, we begin with the K3 surface S/Q whose Néron-Severi group has rank 20 and discriminant −163 and consists of divisor classes defined over Q. 3 We use models of S as the double cover W 2 = P 6 (X, Y, Z) of P 2 branched along a sextic curve C 6 : P 6 (X, Y, Z) = 0. There is a finite but large number (50+) of lines l i : λ i (X, Y, Z) = 0 on which P 6 restricts to a perfect square (geometrically these are the tritangent lines of C 6 ). The restriction P 6 | L of P 6 to a generic line L ⊂ P 2 thus yields a genus-2 curve C L : w 2 = P 6 | L with a pair of rational points above the intersection of L with each l i .2 It is essential to use N (g, K) here rather than B(g, K), because even for a single curve C over a number field K 0 it is clear that by enlarging K ⊃ K 0 we can make #C(K) arbitrarily large.3 See [E1, p.9] for a model of S as an elliptic K3 surface with Mordell-Weil group (Z/4Z) ⊕ Z 4 . 53