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**1 - 1**of**1**### Once-punctured Elliptic Curves

, 2014

"... Abstract. — In the present paper, we prove the finiteness of the set of moderate rational points of a once-punctured elliptic curve over a number field. This finiteness may be regarded as an analogue for a once-punctured elliptic curve of the well-known finiteness of the set of torsion rational poin ..."

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Abstract. — In the present paper, we prove the finiteness of the set of moderate rational points of a once-punctured elliptic curve over a number field. This finiteness may be regarded as an analogue for a once-punctured elliptic curve of the well-known finiteness of the set of torsion rational points of an abelian variety over a number field. In order to obtain the finiteness, we discuss the 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 odd, a necessary and sufficient condition for a certain hyperbolic curve over a generalized sub-l-adic field to have trivial center.