Dyadic torsion of 2-dimensional hyperelliptic Jacobians

TL;DR

This paper constructs a tower of field extensions related to the 2-power torsion points of a hyperelliptic Jacobian over a transcendental field, revealing the Galois structure of these extensions.

Contribution

It explicitly describes the Galois subgroup and generators of the field extension generated by all 2-power torsion points of a hyperelliptic Jacobian, using recursive formulas.

Findings

Identifies the subextension of the 2-power torsion field corresponding to a central Galois subgroup.

Provides recursive formulas for generators of the extension tower.

Shows the explicit generator of the full 2-power torsion field over the constructed tower.

Abstract

Let $k$ be a field of characteristic $0$, and let $\alpha_{1}$, $\alpha_{2}$, ..., $\alpha_{5}$ be algebraically independent and transcendental over $k$. Let $K$ be the transcendental extension of $k$ obtained by adjoining the elementary symmetric functions of the $\alpha_{i}$'s. Let $J$ be the Jacobian of the hyperelliptic curve defined over $K$ which is given by the equation $y^{2} = \prod_{i = 1}^{5} (x - \alpha_{i})$. We define a tower of field extensions $K = K_{0}' \subset K_{1}' \subset K_{2}' \subset ...$ by giving recursive formulas for the generators of each $K_{n}'$ over $K_{n - 1}'$, and let $K_{\infty}' = \bigcup_{n = 0}^{\infty} K_{n}'$. We show that $K_{\infty}'(\mu_{2})$ is the subextension of the field $K(J[2^{\infty}]) := \bigcup_{n = 0}^{\infty} K(E[2^{n}])$ corresponding to a central order-$2$ Galois subgroup of $\mathrm{Gal}(K(J[2^{\infty}]) / K(\mu_{2}))$, and a generator of $K(J[2^{\infty}])$ over $K_{\infty}'(\mu_{2})$ is given.