Dyadic Torsion of Elliptic Curves

Abstract

Let $k$ be a field of characteristic $0$, and let ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$ 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 $E$ be the elliptic curve defined over $K$ which is given by the equation $y^2=(x-{\alpha }_1)(x-{\alpha }_2)(x-{\alpha }_3)$. 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 $K_{n-1}^{\prime }$. We show that $K_{\infty }^{\prime }$ is a certain central subextension of the field $K(E[2^{\infty }]):={\bigcup }_{n=0}^{\infty }K(E[2^n])$, and a generator of $K(E[2^{\infty }])$ over $K_{\infty }^{\prime }({\mu }_2)$ is given. Moreover, if we assume that $k$ contains all $2$-power roots of unity, for each $n$, we show that $K(E[2^n])$ contains $K_n^{\prime }$ and is contained in a certain quadratic extension of $K_{n+1}^{\prime }$.

This is a revision of Sections 1 and 3 of the last draft of this manuscript; Section 2 was adapted as arXiv:1410.2668

References in corpus (2)

• Revêtements étales et groupe fondamental (SGA 1)
• The abelianization of the level 2 mapping class group