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

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

## Details:

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