|
CM points on products of Drinfeld modular curves
Author(s):
Florian
Breuer
Journal:
Trans. Amer. Math. Soc.
359
(2007),
1351-1374.
MSC (2000):
Primary 11G09;
Secondary 14G35
Posted:
September 19, 2006
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
Let  be a product of Drinfeld modular curves over a general base ring  of odd characteristic. We classify those subvarieties of  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:
-
- 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:
10.1090/S0002-9947-06-04109-2
PII:
S 0002-9947(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
Posted:
September 19, 2006
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org