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

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.