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

  • Jeffrey Yelton EMAIL logo
From the journal Mathematica Slovaca

Abstract

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=i=1d(xαi) , 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(). 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.

MSC 2010: 11G5; 11G10
  1. (Communicated by Filippo Nuccio)

Acknowledgement

The author would like to thank the referee for a number of corrections and suggestions which have improved this text.

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Received: 2018-03-08
Accepted: 2018-06-19
Published Online: 2019-03-18
Published in Print: 2019-04-24

© 2019 Mathematical Institute Slovak Academy of Sciences

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