Cyclic coverings of genus $2$ curves of Sophie Germain type
Forum of Mathematics, Sigma, Tome 12 (2024)
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We consider cyclic unramified coverings of degree d of irreducible complex smooth genus $2$ curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order d. The rich geometry of the associated Prym map has been studied in several papers, and the cases $d=2, 3, 5, 7$ are quite well understood. Nevertheless, very little is known for higher values of d. In this paper, we investigate whether the covering can be reconstructed from its Prym variety, that is, whether the generic Prym Torelli theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for $d\ge 11$ prime such that $\frac {d-1}2$ is also prime. We use results of arithmetic nature on $GL_2$-type abelian varieties combined with theta-duality techniques.
@article{10_1017_fms_2024_42,
author = {J.C. Naranjo and A. Ortega and I. Spelta},
title = {Cyclic coverings of genus $2$ curves of {Sophie} {Germain} type},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {12},
year = {2024},
doi = {10.1017/fms.2024.42},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.42/}
}
TY - JOUR AU - J.C. Naranjo AU - A. Ortega AU - I. Spelta TI - Cyclic coverings of genus $2$ curves of Sophie Germain type JO - Forum of Mathematics, Sigma PY - 2024 VL - 12 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.42/ DO - 10.1017/fms.2024.42 LA - en ID - 10_1017_fms_2024_42 ER -
J.C. Naranjo; A. Ortega; I. Spelta. Cyclic coverings of genus $2$ curves of Sophie Germain type. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.42
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