## Osaka Journal of Mathematics

### On the kernels of the pro-$l$ outer Galois representations associated to hyperbolic curves over number fields

Yuichiro Hoshi

#### Abstract

In the present paper, we discuss the relationship between the Galois extension corresponding to the kernel of the pro-$l$ outer Galois representation associated to a hyperbolic curve over a number field and $l$-moderate points of the hyperbolic curve. In particular, we prove that, for a certain hyperbolic curve, the Galois extension under consideration is generated by the coordinates of the $l$-moderate points of the hyperbolic curve. This may be regarded as an analogue of the fact that the Galois extension corresponding to the kernel of the $l$-adic Galois representation associated to an abelian variety is generated by the coordinates of the torsion points of the abelian variety of $l$-power order. Moreover, we discuss an application of the argument of the present paper to the study of the Fermat equation.

#### Article information

Source
Osaka J. Math., Volume 52, Number 3 (2015), 647-677.

Dates
First available in Project Euclid: 17 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1437137613

Mathematical Reviews number (MathSciNet)
MR3370470

Zentralblatt MATH identifier
06502589

#### Citation

Hoshi, Yuichiro. On the kernels of the pro-$l$ outer Galois representations associated to hyperbolic curves over number fields. Osaka J. Math. 52 (2015), no. 3, 647--677. https://projecteuclid.org/euclid.ojm/1437137613

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