On the elliptic Stark conjecture at primes of multiplicative reduction
In [DLR], Darmon, Lauder, and Rotger formulated a p-adic elliptic Stark conjecture for the twist of an elliptic curve E/Q by the self-dual tensor product ¿ ¿ of two odd and two-dimensional Artin representations. These authors provided abundant numerical evidence and proved the conjecture in the spec...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2019 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/179487 |
| Acceso en línea: | https://hdl.handle.net/2117/179487 https://dx.doi.org/10.1512/iumj.2019.68.7704 |
| Access Level: | acceso abierto |
| Palabra clave: | Curves Arithmetical algebraic geometry Corbes Geometria algèbrica--Aritmètica Classificació AMS::14 Algebraic geometry::14H Curves Classificació AMS::11 Number theory::11G Arithmetic algebraic geometry (Diophantine geometry) Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi numèrica |
| Sumario: | In [DLR], Darmon, Lauder, and Rotger formulated a p-adic elliptic Stark conjecture for the twist of an elliptic curve E/Q by the self-dual tensor product ¿ ¿ of two odd and two-dimensional Artin representations. These authors provided abundant numerical evidence and proved the conjecture in the special setting where p is a prime of good reduction for E and ¿ and ¿2 are induced from finite-order characters ¿, ¿ of the same imaginary quadratic field. The key step in their proof is a factorization of one-variable p-adic L-functions, where ¿ varies in a p-adic family of Hecke characters. The main goal of this article is to prove a new case of the conjecture, placing ourselves in the setting where p is a prime of multiplicative reduction for E. In order to achieve our theorem, we need to work with two-variable p-adic L-functions, where the weight 2 cusp form associated with E also moves independently along a Hida family. Our main result then follows from a factorization of p-adic L-series extending to two variables the one obtained in [DLR]. On the way we also generalize to our setting the results obtained in [CR]. |
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