Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank

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• 作者：Richard Pink (1)
  
• 关键词：Primary 11F52 ; Secondary 11G09 ; 14J15 ; 14M27
• 刊名：manuscripta mathematica
• 出版年：2013
• 出版时间：4 - March 2013
• 年：2013
• 卷：140
• 期：3
• 页码：333-361
• 全文大小：360KB
• 参考文献：1. Breuer, F.: Special subvarieties of Drinfeld modular varieties. J. Reine. Angew. Math. doi:10.1515/crelle.2011.136 (in press)
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  5. Gekeler, E.-U.: Moduli for drinfeld modules. In: Goss, D., et聽al. (eds.) The Arithmetic of Function Fields. Proceedings of the Workshop at Ohio State University 1991, pp. 153鈥?70. Walter de Gruyter, Berlin (1992)
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  12. Kapranov, M. M.: Cuspidal divisors on the modular varieties of elliptic modules. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 568鈥?83, 688; translation in Math. USSR-Izv. 30 (1988), no. 3, 533鈥?47
  13. Matsumura H.: Commutative Ring Theory Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989)
  14. Pink, R., Schieder, S.: Compactification of a Drinfeld period domain over a finite field. Preprint July 2010, 42p., arXiv:1007.4796v1 [math.AG], to appear J. Algebr. Geom.
  15. Sa茂di, M.: Moduli schemes of Drinfeld modules. In: Drinfeld Modules, Modular Schemes and Applications (Alden-Biesen, 1996), pp 17鈥?1. World Scientific Publishers, River Edge (1997)
  16. Smith L.: Polynomial invariants of finite groups. Research Notes in Mathematics, vol. 6. A K Peters, Wellesley (1995)
  
• 作者单位：Richard Pink (1)
  
  1. Department of Mathematics, ETH Z眉rich, 8092, Z眉rich, Switzerland
  
• ISSN：1432-1785

文摘

We give an abstract characterization of the Satake compactification of a general Drinfeld modular variety. We prove that it exists and is unique up to unique isomorphism, though we do not give an explicit stratification by Drinfeld modular varieties of smaller rank which is also expected. We construct a natural ample invertible sheaf on it, such that the global sections of its k-th power form the space of (algebraic) Drinfeld modular forms of weight k. We show how the Satake compactification and modular forms behave under all natural morphisms between Drinfeld modular varieties; in particular we define Hecke operators. We give explicit results in some special cases.