# Galois theory and torsion points on curves

Matthew H. Baker; Kenneth A. Ribet

Journal de théorie des nombres de Bordeaux (2003)

- Volume: 15, Issue: 1, page 11-32
- ISSN: 1246-7405

## Access Full Article

top## Abstract

top## How to cite

topBaker, Matthew H., and Ribet, Kenneth A.. "Galois theory and torsion points on curves." Journal de théorie des nombres de Bordeaux 15.1 (2003): 11-32. <http://eudml.org/doc/249096>.

@article{Baker2003,

abstract = {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 $X_0( p)$, embedded in its jacobian by a cuspidal embedding, are the cusps (together with the hyperelliptic branch points when $X_0( p)$ 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.},

author = {Baker, Matthew H., Ribet, Kenneth A.},

journal = {Journal de théorie des nombres de Bordeaux},

keywords = {modular curve; torsion point; abelian variety; hyperelliptic curve; cuspidal subgroup},

language = {eng},

number = {1},

pages = {11-32},

publisher = {Université Bordeaux I},

title = {Galois theory and torsion points on curves},

url = {http://eudml.org/doc/249096},

volume = {15},

year = {2003},

}

TY - JOUR

AU - Baker, Matthew H.

AU - Ribet, Kenneth A.

TI - Galois theory and torsion points on curves

JO - Journal de théorie des nombres de Bordeaux

PY - 2003

PB - Université Bordeaux I

VL - 15

IS - 1

SP - 11

EP - 32

AB - 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 $X_0( p)$, embedded in its jacobian by a cuspidal embedding, are the cusps (together with the hyperelliptic branch points when $X_0( p)$ 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.

LA - eng

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

UR - http://eudml.org/doc/249096

ER -

## References

top- [1] M. Baker, Torsion points on modular curves. Ph.D. thesis, University of California, Berkeley, 1999. MR1760749
- [2] M. Baker, Torsion points on modular curves. Invent. Math.140 (2000), 487-509. Zbl0972.11057MR1760749
- [3] M. Baker, B. Poonen, Torsion packets on curves. Compositio Math.127 (2001), 109-116. Zbl0987.14020MR1832989
- [4] A. Buium, Geometry of p-jets. Duke Math. J.82 (1996), 349-367. Zbl0882.14007MR1387233
- [5] F. Calegari, Almost rational torsion points on elliptic curves. International Math. Res. Notices10 (2001), 487-503. Zbl1002.14004MR1832537
- [6] R.F. Coleman, Ramified torsion points on curves. Duke Math J.54 (1987), 615-640. Zbl0626.14022MR899407
- [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. Zbl0978.11027MR1703745
- [8] R.F. Coleman, A. Tamagawa, P. Tzermias, The cuspidal torsion packet on the Fermat curve. J. Reine Angew. Math.496 (1998), 73-81. Zbl0931.11024MR1605810
- [9] J. Csirik, On the kernel of the Eisenstein ideal. J. Number Theory92 (2002), 348-375. Zbl1004.11035MR1884708
- [10] H.M. Farkas, I. Kra, Riemann Surfaces (second edition). Graduate Texts in Mathematics, vol. 71, Springer-Verlag, Berlin and New York, 1992. Zbl0764.30001MR1139765
- [11] A. Grothendieck, SGA7 I, Exposé IX, Lecture Notes in Mathematics, vol. 288, Springer-Verlag, Berlin and New York, 1972, 313-523. MR354656
- [12] M. Hindry, Autour d'une conjecture de Serge Lang. Invent. Math.94 (1988), 575-603. Zbl0638.14026MR969244
- [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. Zbl0151.27401MR190146
- [15] S. Lang, Fundamentals of Diophantine Geometry. Springer-Verlag, Berlin and New York, 1983. Zbl0528.14013MR715605
- [16] B. Mazur, Modular curves and the Eisenstein ideal. Publ. Math. IHES47 (1977), 33-186. Zbl0394.14008MR488287
- [17] B. Mazur, Rational isogenies of prime degree. Invent. Math.44 (1978), 129-162. Zbl0386.14009MR482230
- [18] B. Mazur, P. Swinnerton-Dyer, Arithmetic of Weil curves. Invent. Math.25 (1974), 1-61. Zbl0281.14016MR354674
- [19] M. Mcquillan, Division points on semi-abelian varieties. Invent. Math.120 (1995), 143-159. Zbl0848.14022MR1323985
- [20] A.P. Ogg, Hyperelliptic modular curves. Bull. Soc. Math. France102 (1974), 449-462. Zbl0314.10018MR364259
- [21] B. Poonen, Mordell-Lang plus Bogomolov. Invent. Math.137 (1999), 413-425. Zbl0995.11040MR1705838
- [22] B. Poonen, Computing torsion points on curves, Experimental Math.10 (2001), no. 3, 449-465. Zbl1063.11017MR1917430
- [23] M. Raynaud, Courbes sur une variété abélienne et points de torsion. Invent. Math.71 (1983), 207-233. Zbl0564.14020MR688265
- [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. Zbl0581.14031
- [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. Zbl1013.11024MR1754687
- [26] K. Ribet, On modular representations of Gal(Q/Q) arising from modular forms. Invent. Math.100 (1990), 431-476. Zbl0773.11039MR1047143
- [27] D.E. Rohrlich, Points at infinity on the Fermat curves. Invent. Math.39 (1977), 95-127. Zbl0357.14010MR441978
- [28] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(Q/Q). Duke Math. J.54 (1987), 179-230. Zbl0641.10026MR885783
- [29] A. Tamagawa, Ramified torsion points on curves with ordinary semistable Jacobian varieties. Duke Math. J.106 (2001), 281-319. Zbl1010.14007MR1813433
- [30] J.-P. Wintenberger, Démonstration d'une conjecture de Lang dans des cas particuliers, preprint.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.