Jacobians of etale covers of the projective line minus three points

Abstract

We consider the outer pro-2 Galois representation on the algebraic fundamental group of the projective line minus three points. This representation has a kernel, whose fixed field Ω₂, is a pro-2 extension of Q(μ₂∞), unramified away from 2. The fields of 2-power torsion of the Jacobians of curves defined over Q, possessing good reduction away from 2, are also pro-2 extensions of Q(μ₂∞), unramified away from 2. In this dissertation, we show that these fields are contained in O2 for certain choices of curves. In particular, the result is shown for all elliptic curves over Q with good reduction away from 2. In proving this theorem, we will demonstrate that these curves appear in the tower of finite etale 2-covers of the projective line minus three points. In the final chapter, we briefly consider three natural generalizations of the result and give partial results in these cases. Specifically, we consider the case of elliptic curves defined over certain extensions of Q, the case of the prime ℓ = 3, and the case of higher genus curves occurring as 2-covers.