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Baker, Matthew H.; Ribet, Kenneth A.

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

J. Théor. Nombres Bordx. 15, No. 1, 11-32 (2003).
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.
Reviewer: D. Poulakis (Thessaloniki)

Cited in 8 Documents

MSC:

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

Keywords:

modular curve; torsion point; abelian variety; hyperelliptic curve; cuspidal subgroup

Citations:

Zbl 0564.14020; Zbl 0626.14022
Cite Review PDF
Full Text: DOI arXiv Numdam EuDML

References:

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