Galois theory and torsion points on curves. (Q1424567)

Let \(K\) be a number field, \(\overline K\) an algebraic closure for \(K\) and \(X\) an algebraic curve over \(K\) of genus \(g\geq 2\). Assume that \(X\) is embedded in its Jacobian \(J\) via a \(K\)-rational Albanese map \(i: X\to J\). The Manin-Mumford conjecture states that the set \(X(\overline K)\cap J(\overline K)^{\text{tors}}\) is finite. The first proof of this conjecture was provided by \textit{M. Raynaud} [Invent. Math. 74, 207--233 (1983; Zbl 0564.14020) and some years later a second by \textit{R. F. Coleman} [Duke Math J. 541, 615--640 (1987; Zbl 0626.14022)]. In case where \(X= X_0(p)\), with \(p\) a prime \(\geq 23\), the Coleman-Kaskel-Ribet conjecture states that the set of torsion points on \(X\) in the embedding \(i_\infty:X\to J\) is precisely \(\{0,\infty\}\cup H\), where \(H\) is the set of hyperelliptic branch points on \(X\) when \(X\) is hyperelliptic and \(p\neq 37\) and \(H=\varnothing\) otherwise. In the paper under review, the authors deal with Galois-theoretic techniques for studying torsion points on curves and give new proofs of the above results.