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Odile Lecacheux

Publications and source records attributed to Odile Lecacheux.

7 recordsLinked to original sources
arXiv2024

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

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.

math.AG↗
arXiv2022

Transcendental lattices of certain singular K3 surfaces

We compute the transcendental lattices of the singular K3 surfaces belonging to three pencils of K3 surfaces, namely the Apéry-Fermi pencil with transcendental lattice $U\oplus \langle 12 \rangle$, the Verrill's pencil with transcendental lattice $U \oplus \langle 6 \rangle$ and another pencil linked to Verrill's pencil with transcendental lattice $U \oplus \langle 24 \rangle$. Many corollaries are deduced. For example, some singular K3 surfaces belong to different pencils or are Kummer surfaces of K3 surfaces of another pencil.

math.AG↗
arXiv2020

Some observations about isogenies between $K3$ surfaces

Even if there are too many elliptic fibrations to investigate and describe on the singular $K3$ surface $Y_{10}$ of discriminant 72 and belonging to the Apéry-Fermi pencil $(Y_k)$, we find on it many interesting properties. For example some of its elliptic fibrations with 3-torsion section induce by 3-isogeny either an elliptic fibration of $Y_2$, the unique $K3$ surface of discriminant 8, or an elliptic fibration of other $K3$ surfaces of discriminant 72.

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arXiv2018

Apéry-Fermi pencil of $K3$-surfaces and their $2$-isogenies

Given a generic $K3$ surface $Y_k$ of the Apéry-Fermi pencil, we use the Kneser-Nishiyama technique to determine all its non isomorphic elliptic fibrations. These computations lead to determine those fibrations with 2-torsion sections T. We classify the fibrations such that the translation by T gives a Shioda-Inose structure. The other fibrations correspond to a K3 surface identified by it transcendental lattice. The same problem is solved for a singular member $Y_2$ of the family showing the differences with the generic case. In conclusion we put our results in the context of relations between $2$-isogenies and isometries on the singular surfaces of the family.

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arXiv2016

Automorphisms of certain Niemeier lattices and Elliptic Fibrations

Nishiyama introduced a lattice theoretic classification of the elliptic fibrations on a $K3$ surface. In a previous paper we used his method to exhibit $52$ elliptic fibrations, up to isomorphisms, of the singular $K3$ surface of discriminant $-12$. We prove here that the list is complete with a $53$th fibration, thanks to a remark of Elkies and Schütt. We characterize the fibration both theoretically and with a Weierstrass model.

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