Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-06-04109-2
Published electronically: September 19, 2006
MathSciNet review: 2262854
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. Y. André, ``Finitude des couples d'invariants modulaires singuliers sur une courbe algébrique plane non modulaire'', J. Reine Angew. Math. 505 (1998), 203-208.MR 1662256 (2000a:11090)
  • 2. F. Breuer, ``Higher Heegner points on elliptic curves over function fields'', J. Number Theory 104 (2004), 315-326. MR 2029509 (2005g:11092)
  • 3. F. Breuer, ``The André-Oort conjecture for products of Drinfeld modular curves'', J. Reine Angew. Math. 579 (2005), 115-144. MR 2124020 (2005k:11113)
  • 4. S.J. Edixhoven, ``Special points on products of modular curves'', Duke Math. J. 126 (2005), no. 2, 325-348. MR 2115260
  • 5. M. Fried and M. Jarden, ``Field Arithmetic'', Springer-Verlag, 1986.MR 0868860 (89b:12010)
  • 6. W. Fulton, ``Intersection Theory'', Springer-Verlag, 1984. MR 0732620 (85k:14004)
  • 7. E.-U. Gekeler, ``Drinfeld Modular Curves'', Lecture Notes in Mathematics 1231, Springer-Verlag, 1986. MR 0874338 (88b:11077)
  • 8. E.-U. Gekeler and M. Reversat, ``Jacobians of Drinfeld modular curves'', J. Reine Angew. Math. 476 (1996), 27-93.MR 1401696 (97f:11043)
  • 9. D. Goss, ``Basic Structures of Function Field Arithmetic'', Springer-Verlag, 1996.MR 1423131 (97i:11062)
  • 10. D. Hayes, ``Explicit class field theory in global function fields'', in: ``Studies in algebra and number theory'' (G.C.Rota, ed.), Academic Press, New York, 1979.MR 0535766 (81d:12011)
  • 11. D. Hayes, ``A Brief introduction to Drinfeld modules'', in: The Arithmetic of Function Fields (eds. D. Goss et al.), de Gruyter, New York, Berlin, 1992.MR 1196509 (93m:11050)
  • 12. L.K. Hua, appendix to: J. Dieudonné, ``On the automorphisms of the classical groups'', Memoirs Amer. Math. Soc. 2 (1951), 1-95. MR 0606555 (82c:20079)
  • 13. G.-J. van der Heiden, ``Weil pairing and the Drinfeld modular curve'', Ph.D. thesis, Rijksuniversiteit Groningen, 2003.
  • 14. W. Lütkebohmert, ``Der Satz von Remmert-Stein in der nichtarchimedischen Funktionentheorie'', Math. Z. 139 (1974), 69-84.MR 0352527 (50:5014)
  • 15. J. Neukirch, ``Algebraische Zahlentheorie'', Springer-Verlag, 1992.MR 1697859 (2000m:11104)
  • 16. J.-P. Serre, ``Trees'', Springer-Verlag, 1980. MR 0607504 (82c:20083)
  • 17. H. Stichtenoth, ``Algebraic Function Fields and Codes'', Springer-Verlag, 1993.MR 1251961 (94k:14016)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11G09, 14G35

Retrieve articles in all journals with MSC (2000): 11G09, 14G35


Additional Information

Florian Breuer
Affiliation: Department of Mathematical Sciences, University of Stellenbosch, Stellenbosch, 7600, South Africa
Email: fbreuer@sun.ac.za

DOI: https://doi.org/10.1090/S0002-9947-06-04109-2
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.

American Mathematical Society