Isogenies between $K3$ surfaces of the Apéry-Fermi pencil

Abstract

Elliptic fibrations of $K3$ surfaces belonging to the Apéry-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 $[4018]$ or $\left[2036\right]$. We also explicitly link $3$-isogeny on a fibration and base change on other fibrations.

39 pages. The difference with the earlier versions is that certain results have been proved with new methods. arXiv admin note: substantial text overlap with arXiv:2006.16108