Dyadic torsion of 2-dimensional hyperelliptic Jacobians

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 $K=K_0^{\prime }\subset K_1^{\prime }\subset K_2^{\prime }\subset ...$ by giving recursive formulas for the generators of each $K_n^{\prime }$ over $K_{n-1}^{\prime }$, and let $K_{\infty }^{\prime }={\bigcup }_{n=0}^{\infty }{K}_n^{\prime }$. We show that $K_{\infty }^{\prime }({\mu }_2)$ 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 $K_{\infty }^{\prime }({\mu }_2)$ is given.

18 pages