Serre’s Constant of Elliptic Curves Over the Rationals
Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre’s constant associated to E, that gives necessary conditions to conclude that (Formula presented.) the mod m Galois representa...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/710725 |
| Acceso en línea: | http://hdl.handle.net/10486/710725 https://dx.doi.org/10.1080/10586458.2019.1655816 |
| Access Level: | acceso abierto |
| Palabra clave: | elliptic curves Galois representation rationals Matemáticas |
| Sumario: | Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre’s constant associated to E, that gives necessary conditions to conclude that (Formula presented.) the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying (Formula presented.) then (Formula presented.) is non-surjective. Conditionally under Serre’s Uniformity Conjecture, we determine all the Serre’s constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod p Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over (Formula presented.) and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of (Formula presented.) Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre’s constants |
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