## Tokyo Journal of Mathematics

### A Construction of Everywhere Good $\mathbf{Q}$-Curves with $p$-Isogeny

Atsuki UMEGAKI

#### Abstract

An elliptic curve $E$ defined over $\bar{\mathbf{Q}}$ is called a $\mathbf{Q}$-curve, if $E$ and $E^\sigma$ are isogenous over $\bar{\mathbf{Q}}$ for any $\sigma$ in $\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})$. For a real quadratic field $K$ and a prime number $p$, we consider a $\mathbf{Q}$-curve $E$ with the following properties: 1) $E$ is defined over $K$, 2) $E$ has everywhere good reduction over $K$, 3) there exists a $p$-isogeny between $E$ and its conjugate $E^\sigma$. In this paper, a method to construct such a $\mathbf{Q}$-curve $E$ for some $p$ will be given.

#### Article information

Source
Tokyo J. Math., Volume 21, Number 1 (1998), 183-200.

Dates
First available in Project Euclid: 31 March 2010

https://projecteuclid.org/euclid.tjm/1270041995

Digital Object Identifier
doi:10.3836/tjm/1270041995

Mathematical Reviews number (MathSciNet)
MR1630171

Zentralblatt MATH identifier
0922.14021

#### Citation

UMEGAKI, Atsuki. A Construction of Everywhere Good $\mathbf{Q}$-Curves with $p$-Isogeny. Tokyo J. Math. 21 (1998), no. 1, 183--200. doi:10.3836/tjm/1270041995. https://projecteuclid.org/euclid.tjm/1270041995