Birch’s lemma over global function fields
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- by Yi Ouyang and Shenxing Zhang PDF
- Proc. Amer. Math. Soc. 145 (2017), 577-584 Request permission
Abstract:
We obtain a function field version of Birch’s Lemma, which reveals non-torsion points in quadratic twists of an elliptic curve over a global function field, where the quadratic twists have many prime factors. The proof uses Brown’s Euler system of Heegner points over function fields and a result of Vigni on the ring class eigenspaces of Mordell-Weil groups in positive characteristic.References
- M. L. Brown, On a conjecture of Tate for elliptic surfaces over finite fields, Proc. London Math. Soc. (3) 69 (1994), no. 3, 489–514. MR 1289861, DOI 10.1112/plms/s3-69.3.489
- M. L. Brown, Heegner modules and elliptic curves, Lecture Notes in Mathematics, vol. 1849, Springer-Verlag, Berlin, 2004. MR 2082815, DOI 10.1007/b98488
- John Coates, Yongxiong Li, Ye Tian, and Shuai Zhai, Quadratic twists of elliptic curves, Proc. Lond. Math. Soc. (3) 110 (2015), no. 2, 357–394. MR 3335282, DOI 10.1112/plms/pdu059
- Ernst-Ulrich Gekeler, Drinfel′d modular curves, Lecture Notes in Mathematics, vol. 1231, Springer-Verlag, Berlin, 1986. MR 874338, DOI 10.1007/BFb0072692
- E.-U. Gekeler and M. Reversat, Jacobians of Drinfeld modular curves, J. Reine Angew. Math. 476 (1996), 27–93. MR 1401696, DOI 10.1515/crll.1996.476.27
- David R. Hayes, Explicit class field theory in global function fields, Studies in algebra and number theory, Adv. in Math. Suppl. Stud., vol. 6, Academic Press, New York-London, 1979, pp. 173–217. MR 535766
- Mihran Papikian, On the degree of modular parametrizations over function fields, J. Number Theory 97 (2002), no. 2, 317–349. MR 1942964, DOI 10.1016/S0022-314X(02)00016-1
- Andreas Schweizer, Hyperelliptic Drinfeld modular curves, Drinfeld modules, modular schemes and applications (Alden-Biesen, 1996) World Sci. Publ., River Edge, NJ, 1997, pp. 330–343. MR 1630612
- John Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 306, 415–440. MR 1610977
- Douglas Ulmer, Elliptic curves and analogies between number fields and function fields, Heegner points and Rankin $L$-series, Math. Sci. Res. Inst. Publ., vol. 49, Cambridge Univ. Press, Cambridge, 2004, pp. 285–315. MR 2083216, DOI 10.1017/CBO9780511756375.011
- Stefano Vigni, On ring class eigenspaces of Mordell-Weil groups of elliptic curves over global function fields, J. Number Theory 128 (2008), no. 7, 2159–2184. MR 2423756, DOI 10.1016/j.jnt.2007.11.007
- Fu-Tsun Wei and Jing Yu, On the independence of Heegner points in the function field case, J. Number Theory 130 (2010), no. 11, 2542–2560. MR 2678861, DOI 10.1016/j.jnt.2010.05.006
Additional Information
- Yi Ouyang
- Affiliation: Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
- Email: yiouyang@ustc.edu.cn
- Shenxing Zhang
- Affiliation: Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China – and – Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
- Email: zsxqq@mail.ustc.edu.cn
- Received by editor(s): December 16, 2015
- Received by editor(s) in revised form: April 21, 2016, and April 30, 2016
- Published electronically: October 24, 2016
- Communicated by: Romyar T. Sharifi
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 577-584
- MSC (2010): Primary 11G05; Secondary 11D25, 11G40
- DOI: https://doi.org/10.1090/proc/13265
- MathSciNet review: 3577862