Isogenies between $K3$ surfaces of the Ap\'ery-Fermi pencil

TL;DR

This paper explores isogenies between specific K3 surfaces in the Apéry-Fermi pencil, revealing their automorphisms, quotient surfaces, and links to complex multiplication, with detailed results on their transcendental lattices.

Contribution

It provides new explicit descriptions of isogenies and automorphisms of K3 surfaces in the Apéry-Fermi pencil, connecting them to complex multiplication and lattice structures.

Findings

Quotients of Y_{10} yield Kummer surfaces or Y_{10} itself.

Y_{2} quotients with 3-torsion sections are isomorphic to Y_{10}.

Different quotient surfaces for Y_{10} involve specific transcendental lattices.

Abstract

Elliptic fibrations of $K3$ surfaces belonging to the Ap\'ery-Fermi pencil ($Y_k$) may have $2$ or $3$-torsion sections defining on $(Y_k)$ automorphisms $\tau$ of order $2$ or $3$. First we consider $Y_{k}/\tau$ \ for some fibrations of the singular $K3$ surface $Y_{10}$ in the case of two-torsion sections and obtain as for the singular surface $Y_{2}$ either the Kummer surface associated to $Y_{10}$ or $Y_{10}$ itself. This last case is associated with the complex multiplication on $Y_{10}$. We prove also that for all the fibrations of $Y_{2}$ with $3$-torsion sections $Y_{2}/\tau=Y_{10}.$ Results are different for $Y_{10}$ where we can obtain for $Y_{10}/\tau$ one of the two surfaces with transcendental lattice $[4 \quad 0\quad 18]$ or $\left[2 \quad 0 \quad 36\right]$. We also explicitly link $3$-isogeny on a fibration and base change on other fibrations.