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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.

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
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References:

[1] Baker, M., Torsion points on modular curves. Ph.D. thesis, University of California, Berkeley, 1999.
[2] Baker, M., Torsion points on modular curves. Invent. Math.140 (2000), 487-509. · Zbl 0972.11057
[3] Baker, M., Poonen, B., Torsion packets on curves. Compositio Math.127 (2001), 109-116. · Zbl 0987.14020
[4] Buium, A., Geometry of p-jets. Duke Math. J.82 (1996), 349-367. · Zbl 0882.14007
[5] Calegari, F., Almost rational torsion points on elliptic curves. International Math. Res. Notices10 (2001), 487-503. · Zbl 1002.14004
[6] Coleman, R.F., Ramified torsion points on curves. Duke Math J.54 (1987), 615-640. · Zbl 0626.14022
[7] Coleman, R.F., Kaskel, B., Ribet, K., Torsion points on X0(N). In Proceedings of a Symposia in Pure Mathematics, 66 (Part 1) Amer. Math. Soc., Providence, RI (1999), 27-49. · Zbl 0978.11027
[8] Coleman, R.F., Tamagawa, A., Tzermias, P., The cuspidal torsion packet on the Fermat curve. J. Reine Angew. Math.496 (1998), 73-81. · Zbl 0931.11024
[9] Csirik, J., On the kernel of the Eisenstein ideal. J. Number Theory92 (2002), 348-375. · Zbl 1004.11035
[10] Farkas, H.M., Kra, I., Riemann Surfaces (second edition). Graduate Texts in Mathematics, vol. 71, Springer-Verlag, Berlin and New York, 1992. · Zbl 0764.30001
[11] Grothendieck, A., SGA7 I, Exposé IX, , vol. 288, Springer-Verlag, Berlin and New York, 1972, 313-523.
[12] Hindry, M., Autour d’une conjecture de Serge Lang. Invent. Math.94 (1988), 575-603. · Zbl 0638.14026
[13] Kim, M., Ribet, K., Torsion points on modular curves and Galois theory, preprint.
[14] Lang, S., Division points on curves. Ann. Mat. Pura Appl.70 (1965), 229-234. · Zbl 0151.27401
[15] Lang, S., Fundamentals of Diophantine Geometry. Springer-Verlag, Berlin and New York, 1983. · Zbl 0528.14013
[16] Mazur, B., Modular curves and the Eisenstein ideal. Publ. Math. IHES47 (1977), 33-186. · Zbl 0394.14008
[17] Mazur, B., Rational isogenies of prime degree. Invent. Math.44 (1978), 129-162. · Zbl 0386.14009
[18] Mazur, B., Swinnerton-Dyer, P., Arithmetic of Weil curves. Invent. Math.25 (1974), 1-61. · Zbl 0281.14016
[19] Mcquillan, M., Division points on semi-abelian varieties. Invent. Math.120 (1995), 143-159. · Zbl 0848.14022
[20] Ogg, A.P., Hyperelliptic modular curves. Bull. Soc. Math. France102 (1974), 449-462. · Zbl 0314.10018
[21] Poonen, B., Mordell-Lang plus Bogomolov. Invent. Math.137 (1999), 413-425. · Zbl 0995.11040
[22] Poonen, B., Computing torsion points on curves, Experimental Math.10 (2001), no. 3, 449-465. · Zbl 1063.11017
[23] Raynaud, M., Courbes sur une variété abélienne et points de torsion. Invent. Math.71 (1983), 207-233. · Zbl 0564.14020
[24] Raynaud, M., 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 0581.14031
[25] Ribet, K., Torsion points on Jo(N) and Galois representations, in “Arithmetic theory of elliptic curves” (Cetraro, 1997), 145-166, 1716, Springer-Verlag, Berlin and New York, 1999. · Zbl 1013.11024
[26] Ribet, K., On modular representations of Gal(Q/Q) arising from modular forms. Invent. Math.100 (1990), 431-476. · Zbl 0773.11039
[27] Rohrlich, D.E., Points at infinity on the Fermat curves. Invent. Math.39 (1977), 95-127. · Zbl 0357.14010
[28] Serre, J.-P., Sur les représentations modulaires de degré 2 de Gal(Q/Q). Duke Math. J.54 (1987), 179-230. · Zbl 0641.10026
[29] Tamagawa, A., Ramified torsion points on curves with ordinary semistable Jacobian varieties. Duke Math. J.106 (2001), 281-319. · Zbl 1010.14007
[30] Wintenberger, J.-P., Démonstration d’une conjecture de Lang dans des cas particuliers, preprint. · Zbl 1048.14008
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