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## Rational points on twists of X-0(63)

Author
Bruin, N.; Fernández, J.; Gonzalez, J.; Lario, J.-C.
Type of activity
Journal article
Journal
Acta arithmetica
Date of publication
2007-04
Volume
126
Number
4
First page
361
Last page
385
DOI
https://doi.org/10.4064/aa126-4-6
Project funding
Curvas de Shimura, Superficies Abelianas y Thetanullwerte
Repository
http://hdl.handle.net/2117/474
URL
https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/acta-arithmetica/all/126/4/82724/rational-points-on-twists-of-x-0-63
Abstract
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 au...
Keywords
Chabauty methods, Elliptic curves, Galois representations, Genus-three curves, Prym varieties, Quadratic Q-curves
Group of research
TN - Number Theory Research Group