Rational points on twists of X0(63)
Let $\varrho\colon G_\mathbb{Q}\longrightarrow PGL_2(\mathbb{F}_p)$ be a Galois representation with cyclotomic determinant, and let $N>1$ be an integer that is square mod $p$. There exist two twisted modular curves $X^+(N,p)_\varrho$ and $X^+(N,p)'_\varrho$\, defined over~$\mathbb{Q}$ whose...
| Autores: | , , , |
|---|---|
| Tipo de documento: | artigo |
| Data de publicação: | 2006 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositório: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglês |
| OAI Identifier: | oai:upcommons.upc.edu:2117/474 |
| Acesso em linha: | https://hdl.handle.net/2117/474 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Number theory Galois representations Elliptic curves Genus-three curves Prym varieties Chabauty methods Quadratic Q-curves Galois, Teoria de Corbes algèbriques Nombres, Teoria dels Classificació AMS::11 Number theory |
| Resumo: | Let $\varrho\colon G_\mathbb{Q}\longrightarrow PGL_2(\mathbb{F}_p)$ be a Galois representation with cyclotomic determinant, and let $N>1$ be an integer that is square mod $p$. There exist two twisted modular curves $X^+(N,p)_\varrho$ and $X^+(N,p)'_\varrho$\, defined over~$\mathbb{Q}$ whose rational points classify the quadratic $\mathbb{Q}$-curves of degree $N$ realizing $\varrho$. The paper focuses on the only genus-three instance: the case $N\!=7,\,p=3$. From an explicit description of the automorphism group of the modular curve $X_0(63)$, it follows that the twisted curves are isomorphic over $\mathbb{Q}$ in this case. We also obtain a plane quartic equation for the twists and then produce the desired $\mathbb{Q}$-curves, provided that the set of rational points on this quartic can be determined. The existence of elliptic quotients and of an unramified double cover $X(7,3)_\varrho$ having a genus-two quotient permits a variety of combinations of covers and Prym-Chabauty methods to determine these rational points. We include two examples where these methods apply. |
|---|