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Some remarks on a theorem of Parent and generalizing Ogg's conjecture

Published online by Cambridge University Press:  01 September 2009

S. KAMIENNY
S. KAMIENNY*
Affiliation:
Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, U.S.A. e-mail: kamienny@usc.edu
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Abstract

We carry out an Eisenstein prime descent to prove the finiteness of the Mordell-Weil group of the Eisenstein quotients of J1(p) for certain values of p that are relevant to torsion in elliptic curves over cubic fields. We then use this to recover some results of Parent. Our methods suggest a possible generalization of Ogg's conjecture for torsion in elliptic curves over number fields.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 2 , September 2009 , pp. 285 - 293
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Abramovich, D.Formal finiteness and the torsion conjecture on elliptic curves. Astérisque No. 228 (1995), 3, 5–17.Google Scholar
[2]Conrad, B., Edixhoven, B. and Stein, W.J 1(p) has connected Fibers. Documenta Math. (2003), 331–408.CrossRefGoogle Scholar
[3]Deligne, P. Formes modulaires et representationa ℓ-adiques. Séminaire Bourbaki 68/69 no. 355. Lecture Notes in Mathematics 179. (Springer, 1971), 136–172.Google Scholar
[4]Deligne, P. and Rapoport, M. Schémas de modules de courbes elliptiques, Lecture Notes in Mathematics 349. (Springer, 1973).CrossRefGoogle Scholar
[5]Faltings, G.Diophantine approximation on abelian varieties. Ann Math. 133 (1991), 549–576.CrossRefGoogle Scholar
[6]Frey, G. Curves with infinitely many points of fixed degree. Preprint (1992), Institut für Experimentelle Mathematik.Google Scholar
[7]Jeon, D. and Kim, C. On the Arithmetic of Certain Modular Curves. arxiv.org/abs/math/0607611.Google Scholar
[8]Kamienny, S. Harvard Ph.D. Thesis (December 1980).Google Scholar
[9]Kamienny, S.Modular curves and unramified extensions of number fields. Compositio Math. 47 (1982), 223–235.Google Scholar
[10]Kamienny, S.On J 1(p) and the Conjecture of Birch and Swinnerton-Dyer. Duke Math. J. 49 (1982), 329–340.CrossRefGoogle Scholar
[11]Kamienny, S.Rational points on the modular curves and abelian varieties. J. Reine Angew. Math. 359 (1985), 174–187.Google Scholar
[12]Kamienny, S.Some remarks on torsion in elliptic curves. Commun. in Alg., 23 (6) (1995), 2167–2169.CrossRefGoogle Scholar
[13]Kamienny, S.Torsion points on elliptic curves and q-coefficients of modular forms. Invent. Math. 109 (1992), 221–229.CrossRefGoogle Scholar
[14]Kamienny, S. Torsion points on elliptic curves and winding homomorphisms. MSRI preprint series #023–94.Google Scholar
[15]Kamienny, S.Torsion points on elliptic curves over all quadratic fields. Duke Math. J. 53 (1986), 157–162.CrossRefGoogle Scholar
[16]Kamienny, S. Torsion points on elliptic curves over fields of higher degree. IMRN No. 6 (1992), 129–133.Google Scholar
[17]Kamienny, S. and Mazur, B.Rational torsion of prime order in elliptic curves over number fields. Astérisque 228 (1995), 81–100.Google Scholar
[18]Kato, K.p-adic Hodge theory and values of zeta functions of modular forms. Astérisque 295 (2004), 117–290.Google Scholar
[19]Kubert, D.Quadratic relations for generators of units in the modular function field. Math. Ann. 225 (1977), 1–20.CrossRefGoogle Scholar
[20]Kubert, D. and Lang, S.The index of stickelberger ideals of order 2 and cuspidal class numbers. Math. Ann. 237 (1978), 213–232.CrossRefGoogle Scholar
[21]Kubert, D. and Lang, S.Modular Units, (Springer-Verlag, 1981).CrossRefGoogle Scholar
[22]Mazur, B.Modular curves and the eisenstein ideal. Publ. Math. Inst. Hanks Études Sci. 47 (1978), 33–186.CrossRefGoogle Scholar
[23]Mazur, B.On the arithmetic of special values of L-functions. Invent. Math. 55 (1979), 207–240.CrossRefGoogle Scholar
[24]Mazur, B.Rational isogenies of prime degree. Invent. Math, 44 (1978), 129–162.CrossRefGoogle Scholar
[25]Mazur, B. and Tate, J.Points of order 13 on ellitic curves. Invent, Math. 22 (1973), 41–49.CrossRefGoogle Scholar
[26]Mazur, B. and Wiles, A.Class fields of abelian extensions of ℚ. Invent. Math. 76 (1984). 179–330.CrossRefGoogle Scholar
[27]Merel, L.Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124 (1996), 437–449.CrossRefGoogle Scholar
[28]Osterlé, J. Torsion des courbes elliptiques sur les corps de nombres, unpublished manuscript.Google Scholar
[29]Ogg, A.Hyperelliptic modular curves. Bull. Soc. Math. France 102 (1974), 449–462.CrossRefGoogle Scholar
[30]Ogg, A.Rational points of finite order on elliptic curves. Invent. Math. 12 (1971), 105–111.CrossRefGoogle Scholar
[31]Oort, F. and Tate, J.Group schemes of prime order. Ann. Sci. École Norm. Sup. série 4, 3 (1970), 1–21.Google Scholar
[32]Parent, P.No 17-torsion on elliptic curves over cubic number fields. J. Th. Nombres Bordeux, 15. no. 3 (2003), 831–838.CrossRefGoogle Scholar
[33]Parent, P.Torsion des courbes elliptiques sur les corps cubiques. Ann. Inst. Fourier, 50, 3 (2000), 723–749.CrossRefGoogle Scholar
[34]Shimura, G.Introduction to the arithmetic theory of automorphic functions. Publ. Math. Soc. Japan, no. 11. (1971).Google Scholar
[35]Stein, W. The modular forms database: tables. http://modular.math.washington.edu/Tables/Google Scholar
[36]Stevens, G.The cuspidal group and special values of L-functions. Trans. Amer. Math. Soc. 291 no. 2 (1985), 519–550.Google Scholar
[37]Wiles, A.Modular curves and the class group of ℚ(ζp). Invent. Math 58 (1980), 1–35.CrossRefGoogle Scholar