CM points on products of Drinfeld modular curves

Breuer, Florian

doi:10.1090/S0002-9947-06-04109-2

CM points on products of Drinfeld modular curves
HTML articles powered by AMS MathViewer

by Florian Breuer PDF

Trans. Amer. Math. Soc. 359 (2007), 1351-1374 Request permission

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

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, DOI 10.1515/crll.1998.118
Florian Breuer, Higher Heegner points on elliptic curves over function fields, J. Number Theory 104 (2004), no. 2, 315–326. MR 2029509, DOI 10.1016/j.jnt.2003.09.001
Florian Breuer, The André-Oort conjecture for products of Drinfeld modular curves, J. Reine Angew. Math. 579 (2005), 115–144. MR 2124020, DOI 10.1515/crll.2005.2005.579.115
Bas Edixhoven, Special points on products of modular curves, Duke Math. J. 126 (2005), no. 2, 325–348. MR 2115260, DOI 10.1215/S0012-7094-04-12624-7
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, DOI 10.1007/978-3-662-07216-5
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, DOI 10.1007/978-3-662-02421-8
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 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, DOI 10.1007/978-3-642-61480-4
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
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
Jean Dieudonné, On the automorphisms of the classical groups, Memoirs of the American Mathematical Society, No. 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
G.-J. van der Heiden, “Weil pairing and the Drinfeld modular curve”, Ph.D. thesis, Rijksuniversiteit Groningen, 2003.
Werner Lütkebohmert, Der Satz von Remmert-Stein in der nichtarchimedischen Funktionentheorie, Math. Z. 139 (1974), 69–84 (German). MR 352527, DOI 10.1007/BF01194146
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, DOI 10.1007/978-3-662-03983-0
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7
Henning Stichtenoth, Algebraic function fields and codes, Universitext, Springer-Verlag, Berlin, 1993. MR 1251961