Superelliptic curves with large Galois images
Let $r>2$ and $\ell$ be primes. In this paper we study the $\bmod \ell$ Galois representations attached to curves of the form $y^r=f(x)$ where $f$ is monic and has coefficients belonging to the $r$ th cyclotomic field. We provide conditions on the coefficients (and degree) of $f$ which allow one...
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:dnet:ubarcelona__::cfb294e0c32e354a67a8c54cdd5ab7f0 |
| Acceso en línea: | https://hdl.handle.net/2445/228885 |
| Access Level: | acceso abierto |
| Palabra clave: | Geometria algebraica Corbes el·líptiques Teoria de nombres Algebraic geometry Elliptic curves Number theory |
| Sumario: | Let $r>2$ and $\ell$ be primes. In this paper we study the $\bmod \ell$ Galois representations attached to curves of the form $y^r=f(x)$ where $f$ is monic and has coefficients belonging to the $r$ th cyclotomic field. We provide conditions on the coefficients (and degree) of $f$ which allow one to verify the $\bmod \ell$ image is large outside of a (typically small) finite explicit set of primes. We allow all values of $r$ for which the $r$ th cyclotomic field has odd class number. This appears to be the first explicit result for abelian varieties of dimension greater than two and not of $\mathrm{GL}_2$-type which allows the ground field to have unramified extensions. To determine the exact image we study the "endomorphism character", a certain algebraic Hecke character which generalises the CM character. This is achieved in entirety when $r=3$. To the author's knowledge, this is the first accurate description of the full image in the literature. Finally, we give several examples with genus ranging from 10 to 36. Applications to the Inverse Galois Problem are also included. |
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