Torsion points on modular curves
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'' Galois orbits, large enough that the conjugates of such a point satisfy a relation in \(J\) with only a few nonzero terms. The upshot is that either \(P\) lies in \(C\) or else \(X_0(N)\) admits a low degree map to the projective line \({\mathbb P}^1\). If \(P\) lies in \(C\), then one uses a result of Mazur that the only points on \(X_0(N)\) mapping into \(C\) are the cusps (\(0\) and \(\infty\)). On the other hand, if \(X_0(N)\) admits a low degree map to \({\mathbb P}^1\), one finds that \(X_0(N)\) is hyperelliptic with \(P\) fixed by hyperelliptic involution, or at least one obtains an upper bound on \(N\), since the literature contains explicit lower bounds on the degree of a nonconstant map \(X_0(N) \rightarrow {\mathbb P}^1\), and this bound goes to infinity with \(N\). The first proof handles each small \(N\) (up to \(911\)) by an argument involving the discriminant of the Hecke algebra, computed by a program of W. Stein. The second proof instead uses more information about the action of inertia groups on torsion points on \(J_0(N)\) to reduce the amount of explicit computation required (although some is still needed). We now describe some of the other results proved in the paper. A nonempty subset of geometric points on a curve \(X\) of genus at least \(2\) over \(\mathbb Q\) is called a torsion packet if for some \(Q \in X_0(N)(\overline{\mathbb Q})\) it equals the set of points in \(X_0(N)(\overline{\mathbb Q})\) that map to torsion points on the Jacobian, when \(Q\) is used as basepoint for the Albanese embedding; the torsion packet is called trivial if it contains only \(Q\) itself. Let \(N\) be a prime. The author proves that if \(X_0^+(N)\) has genus at least \(2\), the torsion packet of \(X_0^+(N)\) containing the cusp \(\infty\) is trivial. Also, if \(X_0(N)\) has a nontrivial torsion packet, then \(X_0(N)\) admits a map to \({\mathbb P}^1\) of degree at most \(6\), and the latter implies \(N \leq 311\). Similarly, if \(X_0^+(N)\) has a nontrivial torsion packet, then \(X_0^+(N)\) admits a map to \({\mathbb P}^1\) of degree at most \(4\), and the latter implies \(N \leq 479\). Finally the author mentions an application of these results to lower bounds on Mordell-Weil ranks. A proof of the Coleman-Kaskel-Ribet conjecture was found independently by \textit{A. Tamagawa} [Duke Math J. 106, 281-319 (2001)]. It has elements in common with Baker's second proof. Baker's survey article [Galois theory and torsion points on curves, to appear in the Proceedings of the 2001 Journées Arithmétiques, as a volume of J. Théor. Nombres Bordx.] combines arguments from the various proofs and attempts to give a more self-contained presentation. But still, the reader must be prepared to accept several facts from the arithmetic of modular curves, or else must delve into the cited papers of Mazur, Ribet, and others, where they are proved. Finally, we mention two errata. First, the claim ``The field of definition of this point contains \({\mathbb Q}(\sqrt{-N})\)'' in the first proof of Lemma 3.13 seems to be unjustified; but the second proof of Lemma 3.13 is fine, so this does not affect the truth of the conclusions. Second, the author communicated to me that his Corollary 4.2 should be regarded as unproved, since it relies on Theorem 0.4 of [Duke Math. J. 60, 401-423 (1990; Zbl 0744.14014)], whose proof is incomplete; indeed, the reviewer has found a counterexample to both Corollary 4.2 and Theorem 0.4: take \(X=X_0(37)\) and take \(x\) to be the image of the cusp \(\infty\) under the hyperelliptic involution.
scientific article; zbMATH DE number 1462435