Watkins's conjecture for elliptic curves over function fields
| dc.contributor.author | Caro, Jerson | |
| dc.date.accessioned | 2025-01-20T16:05:42Z | |
| dc.date.available | 2025-01-20T16:05:42Z | |
| dc.date.issued | 2024 | |
| dc.description.abstract | In 2002 Watkins conjectured that given an elliptic curve defined over Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {Q}}}$$\end{document}, its Mordell-Weil rank is at most the 2-adic valuation of its modular degree. We consider the analogous problem over function fields of positive characteristic, and we prove it in several cases. More precisely, every modular semi-stable elliptic curve over Fq(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_q(T)$$\end{document} after extending constant scalars, and every quadratic twist of a modular elliptic curve over Fq(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_q(T)$$\end{document} by a polynomial with sufficiently many prime factors satisfy the analogue of Watkins's conjecture. Furthermore, for a well-known family of elliptic curves with unbounded rank due to Ulmer, we prove the analogue of Watkins's conjecture. | |
| dc.description.funder | ANID Doctorado Nacional | |
| dc.fuente.origen | WOS | |
| dc.identifier.doi | 10.1007/s00209-024-03602-9 | |
| dc.identifier.eissn | 1432-1823 | |
| dc.identifier.issn | 0025-5874 | |
| dc.identifier.uri | https://doi.org/10.1007/s00209-024-03602-9 | |
| dc.identifier.uri | https://repositorio.uc.cl/handle/11534/89916 | |
| dc.identifier.wosid | WOS:001338129200006 | |
| dc.issue.numero | 3 | |
| dc.language.iso | en | |
| dc.revista | Mathematische zeitschrift | |
| dc.rights | acceso restringido | |
| dc.subject | Watkins's conjecture | |
| dc.subject | Elliptic curves over function fields | |
| dc.subject | Modular degree | |
| dc.subject | Rank | |
| dc.title | Watkins's conjecture for elliptic curves over function fields | |
| dc.type | artículo | |
| dc.volumen | 308 | |
| sipa.index | WOS | |
| sipa.trazabilidad | WOS;2025-01-12 |