## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 45, Number 2 (2016), 271-291.

### Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves

#### Abstract

In the present paper, we prove the {\it finiteness} of the set of {\it moderate} rational points of a once-punctured elliptic curve over a number field. This {\it finiteness} may be regarded as an analogue for a once-punctured elliptic curve of the well-known {\it finiteness} of the set of torsion rational points of an abelian variety over a number field. In order to obtain the {\it finiteness}, we discuss the {\it center} of the image of the pro-$l$ outer Galois action associated to a hyperbolic curve. In particular, we give, under the assumption that $l$ is {\it odd}, a {\it necessary and sufficient condition} for a certain hyperbolic curve over a generalized sub-$l$-adic field to have {\it trivial center}.

#### Article information

**Source**

Hokkaido Math. J., Volume 45, Number 2 (2016), 271-291.

**Dates**

First available in Project Euclid: 2 August 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1470139405

**Digital Object Identifier**

doi:10.14492/hokmj/1470139405

**Mathematical Reviews number (MathSciNet)**

MR3532133

**Zentralblatt MATH identifier**

06598415

**Subjects**

Primary: 14H30: Coverings, fundamental group [See also 14E20, 14F35]

**Keywords**

moderate point once-punctured elliptic curve hyperbolic curve Galois-like automorphism

#### Citation

HOSHI, Yuichiro. Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves. Hokkaido Math. J. 45 (2016), no. 2, 271--291. doi:10.14492/hokmj/1470139405. https://projecteuclid.org/euclid.hokmj/1470139405