Explicit coverings of families of elliptic surfaces by squares of curves

TL;DR

The paper constructs explicit coverings of elliptic surfaces and K3 surfaces by squares of curves, establishing new links between their Hodge structures, motives, and moduli spaces, and proving cases of the Hodge conjecture.

Contribution

It introduces explicit coverings of elliptic and K3 surfaces by squares of curves, and proves the Hodge conjecture for these surfaces' squares.

Findings

Existence of elliptic surfaces covered by squares of genus 2n+1 curves.

Construction of a correspondence between genus 7 curves and certain K3 surfaces.

Proof of the Hodge conjecture for the squares of these K3 surfaces.

Abstract

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 ${\mathbb Q}(\sqrt{-n})$. 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 ${\mathbb P}^4$ 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.