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

Abstract

Given a field $k$ of characteristic different from $2$ and an integer $d\ge 3$, let $J$ be the Jacobian of the "generic" hyperelliptic curve given by $y^2={\prod }_{i=1}^d(x-{\alpha }_i)$, where the ${\alpha }_i$'s are transcendental and independent over $k$; it is defined over the transcendental extension $K/k$ generated by the symmetric functions of the ${\alpha }_i$'s. We investigate certain subfields of the field $K_{\infty }$ obtained by adjoining all points of $2$-power order of $J(\bar{K})$. In particular, we explicitly describe the maximal abelian subextension of $K_{\infty }/K(J[2])$ and show that it is contained in $K(J[8])$ (resp. $K(J[16])$) if $g\ge 2$ (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 $\mathrm{Gal}(K_{\infty }/K)$ on these subfields.

13 pages, 4 sections. This has been updated to version appearing in Mathematica Slovaca, which reflects minor modifications following suggestions of the referee