Galois theory and torsion points on curves. (English) Zbl 1065.11045

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 M. Raynaud [Invent. Math. 74, 207–233 (1983; Zbl 0564.14020) and some years later a second by 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.


11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G18 Arithmetic aspects of modular and Shimura varieties
14H25 Arithmetic ground fields for curves
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