Home Mathematics Rational torsion of generalized Jacobians of modular and Drinfeld modular curves
Article
Licensed
Unlicensed Requires Authentication

Rational torsion of generalized Jacobians of modular and Drinfeld modular curves

  • Fu-Tsun Wei and Takao Yamazaki EMAIL logo
Published/Copyright: January 20, 2019

Abstract

We consider the generalized Jacobian J~ of the modular curve X0(N) of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the -rational torsion points on J~ up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on J~ and its Eisenstein property.


Communicated by Freydoon Shahidi


Award Identifier / Grant number: 105-2115-M-007-018-MY2

Award Identifier / Grant number: 107-2628-M-007-004-MY4

Award Identifier / Grant number: 15K04773

Funding statement: The first author is supported by Ministry of Science and Technology, Taiwan (grant number 105-2115-M-007-018-MY2 and 107-2628-M-007-004-MY4). The second author is supported by Japan Society for the Promotion of Science KAKENHI Grant (grant number 15K04773).

Acknowledgements

We thank the referee for careful reading which helped improving our manuscript.

References

[1] S. Bae, On the modular equation for Drinfel’d modules of rank 2, J. Number Theory 42 (1992), no. 2, 123–133. 10.1016/0022-314X(92)90016-ISearch in Google Scholar

[2] E.-U. Gekeler, A product expansion for the discriminant function of Drinfel’d modules of rank two, J. Number Theory 21 (1985), no. 2, 135–140. 10.1016/0022-314X(85)90046-0Search in Google Scholar

[3] E.-U. Gekeler, On the Drinfeld discriminant function, Compos. Math. 106 (1997), no. 2, 181–202. 10.1023/A:1000169607214Search in Google Scholar

[4] N. M. Katz, Galois properties of torsion points on abelian varieties, Invent. Math. 62 (1981), no. 3, 481–502. 10.1007/BF01394256Search in Google Scholar

[5] S. Lang, Elliptic Functions, 2nd ed., Grad. Texts in Math. 112, Springer, New York, 1987. 10.1007/978-1-4612-4752-4Search in Google Scholar

[6] S. Ling, On the 𝐐-rational cuspidal subgroup and the component group of J0(pr), Israel J. Math. 99 (1997), 29–54. 10.1007/BF02760675Search in Google Scholar

[7] D. J. Lorenzini, Torsion points on the modular Jacobian J0(N), Compos. Math. 96 (1995), no. 2, 149–172. Search in Google Scholar

[8] J. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66. 10.1142/9789812830517_0010Search in Google Scholar

[9] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33–186. 10.1007/BF02684339Search in Google Scholar

[10] A. P. Ogg, Diophantine equations and modular forms, Bull. Amer. Math. Soc. 81 (1975), 14–27. 10.1090/S0002-9904-1975-13623-8Search in Google Scholar

[11] M. Ohta, Eisenstein ideals and the rational torsion subgroups of modular Jacobian varieties II, Tokyo J. Math. 37 (2014), no. 2, 273–318. 10.3836/tjm/1422452795Search in Google Scholar

[12] A. Pál, On the torsion of the Mordell-Weil group of the Jacobian of Drinfeld modular curves, Doc. Math. 10 (2005), 131–198. 10.4171/dm/185Search in Google Scholar

[13] M. Papikian and F.-T. Wei, The Eisenstein ideal and Jacquet–Langlands isogeny over function fields, Doc. Math. 20 (2015), 551–629. 10.4171/dm/499Search in Google Scholar

[14] M. Papikian and F.-T. Wei, The rational torsion subgroups of Drinfeld modular Jacobians and Eisenstein pseudo-harmonic cochains, Math. Z. 287 (2017), no. 1–2, 521–546. 10.1007/s00209-016-1835-2Search in Google Scholar

[15] J.-P. Serre, Algebraic Groups and Class Fields, Grad. Texts in Math. 117, Springer, New York, 1988. 10.1007/978-1-4612-1035-1Search in Google Scholar

[16] J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math. 106, Springer, New York, 1986. 10.1007/978-1-4757-1920-8Search in Google Scholar

[17] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Grad. Texts in Math. 151, Springer, New York, 1994. 10.1007/978-1-4612-0851-8Search in Google Scholar

[18] T. Yamazaki and Y. Yang, Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus, Doc. Math. 21 (2016), 1669–1690. 10.4171/dm/x11Search in Google Scholar

Received: 2018-06-13
Revised: 2018-11-28
Published Online: 2019-01-20
Published in Print: 2019-05-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 4.2.2026 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2018-0141/pdf
Scroll to top button