Cyclic coverings of genus 2 curves of Sophie Germain type
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 se...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/222606 |
| Acceso en línea: | https://hdl.handle.net/2445/222606 |
| Access Level: | acceso abierto |
| Palabra clave: | Formes de Jacobi Varietats abelianes Corbes algebraiques Jacobi forms Abelian varieties Algebraic curves |
| Sumario: | 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 $d \geq 11$ prime such that $\frac{d-1}{2}$ is also prime. We use results of arithmetic nature on $G L_2$-type abelian varieties combined with theta-duality techniques. |
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