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Transactions of the American Mathematical Society

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CM points on products of Drinfeld modular curves

Author: Florian Breuer
Journal: Trans. Amer. Math. Soc. 359 (2007), 1351-1374
MSC (2000): Primary 11G09; Secondary 14G35
Published electronically: September 19, 2006
MathSciNet review: 2262854
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Abstract: Let $ X$ be a product of Drinfeld modular curves over a general base ring $ A$ of odd characteristic. We classify those subvarieties of $ X$ which contain a Zariski-dense subset of CM points. This is an analogue of the André-Oort conjecture. As an application, we construct non-trivial families of higher Heegner points on modular elliptic curves over global function fields.

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  • 1. Yves André, Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire, J. Reine Angew. Math. 505 (1998), 203–208 (French, with English summary). MR 1662256, 10.1515/crll.1998.118
  • 2. Florian Breuer, Higher Heegner points on elliptic curves over function fields, J. Number Theory 104 (2004), no. 2, 315–326. MR 2029509, 10.1016/j.jnt.2003.09.001
  • 3. Florian Breuer, The André-Oort conjecture for products of Drinfeld modular curves, J. Reine Angew. Math. 579 (2005), 115–144. MR 2124020, 10.1515/crll.2005.2005.579.115
  • 4. Bas Edixhoven, Special points on products of modular curves, Duke Math. J. 126 (2005), no. 2, 325–348. MR 2115260, 10.1215/S0012-7094-04-12624-7
  • 5. Michael D. Fried and Moshe Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 11, Springer-Verlag, Berlin, 1986. MR 868860
  • 6. William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620
  • 7. Ernst-Ulrich Gekeler, Drinfel′d modular curves, Lecture Notes in Mathematics, vol. 1231, Springer-Verlag, Berlin, 1986. MR 874338
  • 8. E.-U. Gekeler and M. Reversat, Jacobians of Drinfeld modular curves, J. Reine Angew. Math. 476 (1996), 27–93. MR 1401696, 10.1515/crll.1996.476.27
  • 9. David Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer-Verlag, Berlin, 1996. MR 1423131
  • 10. 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
  • 11. David R. Hayes, A brief introduction to Drinfel′d modules, The arithmetic of function fields (Columbus, OH, 1991) Ohio State Univ. Math. Res. Inst. Publ., vol. 2, de Gruyter, Berlin, 1992, pp. 1–32. MR 1196509
  • 12. Jean Dieudonné, On the automorphisms of the classical groups, Memoirs of the American Mathematical Society, vol. 2, American Mathematical Society, Providence, R.I., 1980. With a supplement by Loo Keng Hua [Luo Geng Hua]; Reprint of the 1951 original. MR 606555
  • 13. G.-J. van der Heiden, ``Weil pairing and the Drinfeld modular curve'', Ph.D. thesis, Rijksuniversiteit Groningen, 2003.
  • 14. Werner Lütkebohmert, Der Satz von Remmert-Stein in der nichtarchimedischen Funktionentheorie, Math. Z. 139 (1974), 69–84 (German). MR 0352527
  • 15. Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859
  • 16. Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504
  • 17. Henning Stichtenoth, Algebraic function fields and codes, Universitext, Springer-Verlag, Berlin, 1993. MR 1251961

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Additional Information

Florian Breuer
Affiliation: Department of Mathematical Sciences, University of Stellenbosch, Stellenbosch, 7600, South Africa

Keywords: Drinfeld modular curves, CM points, Andr\'e-Oort conjecture, Heegner points
Received by editor(s): September 20, 2004
Received by editor(s) in revised form: March 1, 2005
Published electronically: September 19, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.