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In: Mathematica Slovaca, vol. 69, no. 2
Jeffrey Yelton

An abelian subextension of the dyadic division field of a hyperelliptic Jacobian

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Year, pages: 2019, 357 - 370
Keywords: hyperelliptic curve, abelian field extension, Galois representation

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Given a field $k$ of characteristic different from $2$ and an integer $d ≥ 3$, let $J$ be the Jacobian of the ``generic" hyperelliptic curve given by $y2 = \prodi = 1d (x - αi)$, where the $αi$'s are transcendental and independent over $k$; it is defined over the transcendental extension $K / k$ generated by the symmetric functions of the $αi$'s. We investigate certain subfields of the field $K$ obtained by adjoining all points of $2$-power order of $J(\bar{K})$. In particular, we explicitly describe the maximal abelian subextension of $K / K(J[2])$ and show that it is contained in $K(J[8])$ (resp. $K(J[16])$) if $g ≥ 2$ (resp. if $g = 1$). On the way we obtain an explicit description of the abelian subextension $K(J[4])$, and we describe the action of a particular automorphism in $\Gal(K / K)$ on these subfields.

How to cite:

ISO 690:
Yelton, J. 2019. An abelian subextension of the dyadic division field of a hyperelliptic Jacobian. In Mathematica Slovaca, vol. 69, no.2, pp. 357-370. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0228

APA:
Yelton, J. (2019). An abelian subextension of the dyadic division field of a hyperelliptic Jacobian. Mathematica Slovaca, 69(2), 357-370. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0228

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Published: 27. 3. 2019