Torsion points on modular curves
type: Item
instance of: scholarly article
title: Torsion points on modular curves
zbMATH Open document ID: 0972.11057
author: Matthew Baker
published in: Inventiones Mathematicae
publication date: 2001-11-12T00:00:00Z
review text: Let \(N\) be a prime number, and let \(X_0(N)\) be the usual modular curve. Assume also that \(N \geq 23\); for prime \(N\) this is equivalent to the genus of \(X_0(N)\) being at least \(2\). Embed \(X_0(N)\) in its Jacobian \(J_0(N)\) using the cusp \(\infty\) as basepoint. \textit{R. Coleman, B. Kaskel} and \textit{K. A. Ribet} [Proc. Symp. Pure Math. 66 (pt. 1), 27-49 (1999; Zbl 0978.11027)] gave a conjectural description of the set \(T\) of points in \(X_0(N)(\overline{\mathbb Q})\) mapping to torsion points of \(J_0(N)\). The main result of the current paper is a proof of this conjecture. In fact, the author presents two proofs, and he applies his methods to several other modular curves and other embeddings as well. The finiteness of \(T\) is a special case of the Manin-Mumford Conjecture proved by \textit{M. Raynaud} [Invent. Math. 71, 207-233 (1983; Zbl 0564.14020)], which applies to any curve of genus at least \(2\) over a field of characteristic zero. For a survey of results connected with this result, see \textit{P. Tzermias} [Bull. Lond. Math. Soc. 32, 641-652 (2000; Zbl 1073.14525)]; for some generalizations, see \textit{B. Poonen} [Invent. Math. 137, 413-425 (1999; Zbl 0995.11040)] and the references cited there. What the author proves for \(X_0(N)\) embedded as above, is that \(T\) consists of the two cusps \(0\) and \(\infty\), except when \(X_0(N)\) is hyperelliptic and \(N \not=37\), in which case \(T\) is the union of \(\{0,\infty\}\) and the fixed points of the hyperelliptic involution. The exception occurs if and only if \(N \in \{23, 29, 31, 41, 47, 59, 71 \}\), which holds if and only if the quotient \(X_0^+(N)\) of \(X_0(N)\) by the Atkin-Lehner involution has genus zero. The proofs build on the work of Coleman, Kaskel, and Ribet. Very roughly, the strategy is as follows. Suppose \(P \in T\). Then all Galois conjugates of \(P\) are in \(T\). But torsion points of \(J\) outside the subgroup \(C\) generated by the cusps tend to have ``large'' Ga…
reviewed by: Bjorn Poonen
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 1462435
zbMATH Keywords: modular curves
zbMATH Keywords: torsion points
zbMATH Keywords: Manin-Mumford Conjecture
zbMATH Keywords: torsion packets
MaRDI profile type: Publication
DOI: 10.1007/S002220000055
description: scientific article; zbMATH DE number 1462435