Rational points on twists of X0(63)
| Autor: | Joan-C. Lario, Julio Fernández, Nils Bruin, Josep González |
|---|---|
| Rok vydání: | 2007 |
| Předmět: | |
| Zdroj: | Acta Arithmetica. 126:361-385 |
| ISSN: | 1730-6264 0065-1036 |
| DOI: | 10.4064/aa126-4-6 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|