2020/09/16 by Colin Ingalls, Adam Logan, Ingalls, Colin +3
Computer Science · Mathematics · #14C30 #14H40 (Secondary) #14J27 #14J28 (Primary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2009.07807
openalex publication_date 2020/09/16 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We show that, for each n>0, there is a family of elliptic surfaces which are covered by the square of a curve of genus 2n+1, and whose Hodge structures have an action by \mathbb Q(√(-n)). By considering the case n=3, we show that one particular family of K3 surfaces are covered by the square of genus 7. Using this, we construct a correspondence between the square of a curve of genus 7 and a general K3 surface in \mathbb P4 with 15 ordinary double points up to isogeny. This gives an explicit proof of the Kuga-Satake-Deligne correspondence for these K3 surfaces and any K3 surfaces isogenous to them, and further, a proof of the Hodge conjecture for the squares of these surfaces. We conclude that the motives of these surfaces are Kimura-finite. Our analysis gives a birational equivalence between a moduli space of curves with additional data and the moduli space of these K3 surfaces with a specific elliptic fibration.