Galois theory and torsion points on curves

Baker, Matthew H.; Ribet, Kenneth A.

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 11-32.

Dans cet article, nous exposons diverses techniques de théorie de Galois qui s’appliquent à l’étude des points de torsion des courbes. En particulier, nous donnons de nouvelles démonstrations de résultats de A. Tamagawa et des auteurs concernant les points de torsion des courbes à “bonne” ou “semi-stable” réduction “ordinaire” en un nombre premier donné. Nous donnons également de nouvelles démonstrations de : (1) la conjecture de Manin-Mumford : il n’y a qu’un nombre fini de points de torsion sur une courbe de genre au moins $2$ plongée dans sa jacobienne par l’application d’Albanese ; et (2) : la conjecture de Coleman-Kaskel-Ribet : pour un nombre premier $p\ge 23$, les seuls points de torsion appartenant à la courbe $X_0\left(p\right)$ plongée dans sa jacobienne par un plongement cuspidal sont les pointes (et les points de branchement hyperelliptique lorsque $X_0\left(p\right)$ est hyperelliptique et $p\ne 37$). Afin de rendre l’exposition aussi utile que possible, nous donnons des références pour tous les résultats sur les courbes modulaires qui interviennent dans notre discussion.

In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with “ordinary good” or “ordinary semistable” reduction at a given prime. We also give new proofs of : (1) the Manin-Mumford conjecture : there are only finitely many torsion points lying on a curve of genus at least $2$ embedded in its jacobian by an Albanese map; and (2) the Coleman-Kaskel-Ribet conjecture : if $p$ is a prime number which is at least $23$, then the only torsion points lying on the curve $X_0\left(p\right)$, embedded in its jacobian by a cuspidal embedding, are the cusps (together with the hyperelliptic branch points when $X_0\left(p\right)$ is hyperelliptic and $p$ is not 37). In an effort to make the exposition as useful as possible, we provide references for all of the facts about modular curves which are needed for our discussion.

[1] M. Baker, Torsion points on modular curves. Ph.D. thesis, University of California, Berkeley, 1999. | MR

[2] M. Baker, Torsion points on modular curves. Invent. Math. 140 (2000), 487-509. | MR | Zbl

[3] M. Baker, B. Poonen, Torsion packets on curves. Compositio Math. 127 (2001), 109-116. | MR | Zbl

[4] A. Buium, Geometry of p-jets. Duke Math. J. 82 (1996), 349-367. | MR | Zbl

[5] F. Calegari, Almost rational torsion points on elliptic curves. International Math. Res. Notices 10 (2001), 487-503. | MR | Zbl

[6] R.F. Coleman, Ramified torsion points on curves. Duke Math J. 54 (1987), 615-640. | MR | Zbl

[7] R.F. Coleman, B. Kaskel, K. Ribet, Torsion points on X0(N). In Proceedings of a Symposia in Pure Mathematics, 66 (Part 1) Amer. Math. Soc., Providence, RI (1999), 27-49. | MR | Zbl

[8] R.F. Coleman, A. Tamagawa, P. Tzermias, The cuspidal torsion packet on the Fermat curve. J. Reine Angew. Math. 496 (1998), 73-81. | MR | Zbl

[9] J. Csirik, On the kernel of the Eisenstein ideal. J. Number Theory 92 (2002), 348-375. | MR | Zbl

[10] H.M. Farkas, I. Kra, Riemann Surfaces (second edition). Graduate Texts in Mathematics, vol. 71, Springer-Verlag, Berlin and New York, 1992. | MR | Zbl

[11] A. Grothendieck, SGA7 I, Exposé IX, Lecture Notes in Mathematics, vol. 288, Springer-Verlag, Berlin and New York, 1972, 313-523. | MR

[12] M. Hindry, Autour d'une conjecture de Serge Lang. Invent. Math. 94 (1988), 575-603. | MR | Zbl

[13] M. Kim, K. Ribet, Torsion points on modular curves and Galois theory, preprint.

[14] S. Lang, Division points on curves. Ann. Mat. Pura Appl. 70 (1965), 229-234. | MR | Zbl

[15] S. Lang, Fundamentals of Diophantine Geometry. Springer-Verlag, Berlin and New York, 1983. | MR | Zbl

[16] B. Mazur, Modular curves and the Eisenstein ideal. Publ. Math. IHES 47 (1977), 33-186. | Numdam | MR | Zbl

[17] B. Mazur, Rational isogenies of prime degree. Invent. Math. 44 (1978), 129-162. | MR | Zbl

[18] B. Mazur, P. Swinnerton-Dyer, Arithmetic of Weil curves. Invent. Math. 25 (1974), 1-61. | MR | Zbl

[19] M. Mcquillan, Division points on semi-abelian varieties. Invent. Math. 120 (1995), 143-159. | MR | Zbl

[20] A.P. Ogg, Hyperelliptic modular curves. Bull. Soc. Math. France 102 (1974), 449-462. | Numdam | MR | Zbl

[21] B. Poonen, Mordell-Lang plus Bogomolov. Invent. Math. 137 (1999), 413-425. | MR | Zbl

[22] B. Poonen, Computing torsion points on curves, Experimental Math. 10 (2001), no. 3, 449-465. | MR | Zbl

[23] M. Raynaud, Courbes sur une variété abélienne et points de torsion. Invent. Math. 71 (1983), 207-233. | MR | Zbl

[24] M. Raynaud, Sous-variétés d'une variété abélienne et points de torsion, in Arithmetic and Geometry, Vol. I, Progr. Math. 35, Birkhäuser, Boston, 1983, 327-352. | Zbl

[25] K. Ribet, Torsion points on Jo(N) and Galois representations, in "Arithmetic theory of elliptic curves" (Cetraro, 1997), 145-166, Lecture Notes in Math. 1716, Springer-Verlag, Berlin and New York, 1999. | MR | Zbl

[26] K. Ribet, On modular representations of Gal(Q/Q) arising from modular forms. Invent. Math. 100 (1990), 431-476. | MR | Zbl

[27] D.E. Rohrlich, Points at infinity on the Fermat curves. Invent. Math. 39 (1977), 95-127. | MR | Zbl

[28] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(Q/Q). Duke Math. J. 54 (1987), 179-230. | MR | Zbl

[29] A. Tamagawa, Ramified torsion points on curves with ordinary semistable Jacobian varieties. Duke Math. J. 106 (2001), 281-319. | MR | Zbl

[30] J.-P. Wintenberger, Démonstration d'une conjecture de Lang dans des cas particuliers, preprint.