Isogenies between K3 surfaces of the Apéry-Fermi pencil

2022/03/08 by Marie José Bertin, Bertin, Marie José, Odile Lecacheux +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #11F23 #11G05 #14J27 #14J28 (Primary) #Algebraic Geometry (math.AG) #Amino Acid Enzymes and Metabolism #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2203.04151

openalex publication_date 2022/03/08 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

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 Y₁₀ in the case of two-torsion sections and obtain as for the singular surface Y₂ either the Kummer surface associated to Y₁₀ or Y₁₀ itself. This last case is associated with the complex multiplication on Y₁₀. We prove also that for all the fibrations of Y₂ with 3-torsion sections Y₂/τ=Y₁₀. Results are different for Y₁₀ where we can obtain for Y₁₀/τ one of the two surfaces with transcendental lattice [4 0 18] or [2 0 36]. We also explicitly link 3-isogeny on a fibration and base change on other fibrations.

Related

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• Some observations about isogenies between K3 surfaces
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