Examples of torsion points on genus two curves

Transactions of the American Mathematical Society

Volumn 352, Issue 10, 2000, Pages 4533-4555

  Boxall, John ^a   Grant, David ^b  

^a UNIVERSITÉ DE CAEN   (France)

^b UNIVERSITY OF COLORADO   (United States)

Author keywords

Curves of genus two; Elliptic curves; Galois representations; Torsion

Indexed keywords

EID: 23044522875     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-00-02368-0     Document Type: Article

References (32)

1. [BoMM-B] J.-B. Bost, J.-F. Mestrc, L. Moret-Bailly, Calculs explicite en genre 2, Astérisque 183 (1990), 69-106. MR 92g:14018b
2. [B] A. Buium, Geometry ofp-jets, Duke Math. Jour. 82 (1996), 349-367. MR 97c:14029
3. [Ça] D. Cantor, On the analogue of the division polynomials for hyperelliptic curves, Grelle 447 (1994), 91-145. MR 94m:11071
4. [CasFl] J. W. S. Casscls, E. V. Flynn, Prolegomena to a Middlebrow Arithmetic of Curves of Genus S, Cambridge University Press, 1996. MR 971:11071
5. [Cl] R. F. Coleman, Torsion points on curves and p-adic Abelian integrals, Annals of Math. 121 (1985), 111-168. MR 86j:14014
6. [C2] R. F. Coleman, Torsion points on Fermât curves, Composition Math. 58 (1986), 191-208. MR 87k:14019
7. [C3] R. F. Coleman, Ramified torsion points on curves, Duke Math. J. 54 (1987), 615-640. MR 89c:14033
8. [CKR] R. F. Coleman, B. Kaskel, K. A. Ribet, Torsion points on Xo(N), in Automorphic Forms, Automorphic Representations, and Arithmetic, Proc. Sympos. Pure Math., vol. 66, part 1, 1999, pp. 27-49. CMP 99:16
9. [CTT] R. F. Coleman, A. Tamagawa, P Tzermias, The cuspidal torsion packet on the Fermât curve, J. Reine Angew. Math. 496 (1998), 73-81. MR 99b:11066
10. [Cr] J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, 1992. MR 93m:11053
11. [DaPh] S. David, P. Philippon, Minorations des hauteurs normalisées de sous-variétés de variétés abéliennes, Cont. Math. 210 (1998), 333-364. MR 98j:11044
12. [D] M. Deuring, Die Zetafunktion einer algebraischen Kurve von Geschlechte Eins, I-IV, Gott. Nach. (1953). MR 15:779d
13. [FK] G. Frey, E. Kani, Curves of genus 2 covering elliptic curves and an arithmetic application, Prog. Math. 89 (1991), 153-175. MR 91k:14014
14. 5 + 1/4, Acta Arith LXXV.4 (1996), 321-337. MR 97b:11083
15. 5 + 1/4 and the conjectures of Stark and Rubin, J. Number Theory 77 (1999), 227-251. CMP 99:16
16. [G3] D. Grant, Units from 3- and 4- torsion on Jacobians of curves of genus 2, Compositio Math. 95 (1994), 311-320. MR 95j:11053
17. [Gr] R. Greenberg, On the Jacobian variety of some algebraic curves, Compositio Math. 42 (1981), 345-359. MR 82j:14036
18. [H] M. Hindry, Autour d'une conjecture de Serge Lang, Invent. Math. 94 (1988), 575-603. MR 89k:11046
19. [I] J. Igusa, Arithmetic variety of moduli for genus two, Annals of Math. 72 (1960), 612-649. MR 22:5637
20. [Ku] R: Kühn, Curves of genus 2 with split Jacobian, Trans. AMS 307 (1988), 41-49. MR 89f: 14027
21. [L] S. Lang, Division points on curves, Ann. Math. Pura Appl. LXX (1965), 229-234. MR 32:7560
22. [P] A. Pillay, Model theory and diophantine geometry, Bull. Amer. Math. Soc. 34 (1997), 405-422. MR 98h:11164a
23. [RI] M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), 207-233. MR 84c:14021
24. [R2] M. Raynaud, Sous-variétés d'une variété abélienne et points de torsion, Prog. Math. 35 (1983), 327-352. MR 85k:14022
25. [SI] J.-P. Serre, Abelian l-adic representations and elliptic curves, Benjamin, New York, 1968. MR 41:8422
26. [S2] J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331. MR 52:8126
27. [Sha] D. Shaulis, Torsion points on thé Jacobian of a hyperelliptic image of a Fermât curve, PhD. Thesis, University of Colorado at Boulder (1998).
28. [ShT] G. Shimura, Y. Taniyama, Complex multiplication of Abelian varieties and its applications to number theory, Publ. Math. Soc. Japan No. 6, 1961. MR 23:A2419
29. [Si] J. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986. MR 87g:11070
30. [St] H. Stark, The Coates-Wiles theorem revisited, Prog. Math. 36 (1982), 349-362. MR 84c:14039
31. [U] E. Ullmo, Positivité et discrétion des points algébriques de courbes, Annals of Math. (2) 147 (1998), 167-179. MR 99e:14031
32. [W] A. Weil, On a certain type of character of the idele class group, Proc. Int. Symp. Algebraic Number Theory, Kyoto (1955), 1-7. MR 18,720e