Year: 2022
Type: preprint
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 $τ$ of order $2$ or $3$. First we consider $Y_{k}/τ$ \ 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 $Y_{2}/τ=Y_{10}.$ Results are different for $Y_{10}$ where we can obtain for $Y_{10}/τ$ 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.
Source: arXiv (Cornell University)
Authors: Bertin, Marie José, Lecacheux, Odile
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4226201020