Integral points on conic log K3 surfaces

Abstract

Adapting a powerful method of Swinnerton-Dyer, we give explicit sufficient conditions for the existence of integral points on certain schemes which are fibered into affine conics. This includes, in particular, cases where the scheme is geometrically a smooth log K3 surface. To the knowledge of the author, this is the first family of log K3 surfaces for which such conditions are established. 2-Descent for quadratic norm 1 toriLet k be a number field, S 0 a finite set of places of k and O S 0 the ring of S 0 -integers of k. In this section we will develop a 2-descent formalism whose goal is to yield sufficient conditions for the existence S 0 -integral points on schemes of the form ax 2 + by 2 = 1 (2.0.1)where a, b are a mutually coprime S 0 -integers. To this end we will study certain Tate-Shafarevich groups associated with 2.0.1, and define suitable Selmer groups to compute them. The main result of this section is Corollary 2.3.2 below, which will be used in §3.1 to find S 0 -integral points on schemes which are fibered into affine conics of type 2.0.1. PreliminariesLet d ∈ O S 0 be a non-zero S 0 -integer and let K = k( √ d) be the associated quadratic extension. We will denote by T 0 the set of places of K lying above S 0 . Let us assume that d satisfies the following condition: Assumption 2.1.1. For every v / ∈ S 0 we have val v (d) ≤ 1 and val v (d) = 1 if v lies above 2.The following useful lemma is by no means novel, but we could not find an explicit reference.Lemma 2.1.2. If Assumption 2.1.1 holds then the ring O T 0 is generated, as an O S 0module, by 1 and d.

Brauer–Manin obstruction for Erdős–Straus surfaces

“…Consider the corresponding algebraic surface derived from (1.1) This is an affine cubic surface, and geometrically a so‐called log K3 surface . Many interesting classical Diophantine equations turn out to concern log K3 surfaces, and their integer points are an active area of research [5, 6, 11–13, 17]. Note that is singular, with the unique singular point lying at the origin.…”