Torsion points on abelian étale coverings of $\textbf {P}^ 1-\{0,1,\infty \}$

Abstract:

Let $X \to {{\mathbf {P}}^1}$ be an Abelian covering of degree $m$ over ${\mathbf {Q}}({\mu _m})$ unbranched outside $0$, $1$ and $\infty$. If the genus of $X$ is greater than $1$ embed $X$ in its Jacobian $J$ in such a way that one of the points above $0$, $1$ or $\infty$ is mapped to the origin. We study the set of torsion points of $J$ which lie on $X$. In particular, we prove that this set is defined over an extension of ${\mathbf {Q}}$ unramified outside $6m$. We also obtain information about the orders of these torsion points.