Borcherds products on unitary groups

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• 作者：Eric Hofmann (1)
  
• 关键词：11F27 ; 11F55 ; 11G18 ; 14G35
• 刊名：Mathematische Annalen
• 出版年：2014
• 出版时间：April 2014
• 年：2014
• 卷：358
• 期：3-4
• 页码：799-832
• 全文大小：364 KB
• 参考文献：1. Baily, Jr., W.L., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains (2). Ann. Math. 84, 442-28 (1966)
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  5. Borcherds, R.E.: Correction to: The Gross-Kohnen-Zagier theorem in higher dimensions [Duke Math. J. 97(2), 1999, pp. 219-33; MR1682249 (2000f:11052)]. Duke Math. J. 105(1), 183-84 (2000) CrossRef
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  8. Bruinier, J.H., Howard, B., Yang, T.: Heights of Kudla-Rapoport divisors and derivatives of L-functions. ArXiv e-prints, 1303.0549 (2013)
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  10. Hofmann, E.: Automorphic Products on Unitary Groups. PhD thesis, TU Darmstadt. http://tuprints.ulb.tu-darmstadt.de/2540/ (2011)
  11. Hofmann, E.: Borcherds products for ${\rm U}(1,1)$ . ArXiv e-prints, 1302.5301 (2013)
  12. Liu, Y.: Arithmetic theta lifting and $L$ -derivatives for unitary groups. I. Algebra Number Theory 5(7), 849-21 (2011)
  13. McGraw, W.J.: The rationality of vector valued modular forms associated with the weil representation. Math. Ann. 326, 105-22 (2003) CrossRef
  
• 作者单位：Eric Hofmann (1)
  
  1. Mathematisches Institut Universit?t Heidelberg, Im Neuenheimer Feld 288, 69120?, Heidelberg, Germany
  
• ISSN：1432-1807

文摘

In the present paper, we provide a construction of the multiplicative Borcherds lift for unitary groups $\mathrm U (1,m)$ , which takes weakly holomorphic elliptic modular forms as input functions and lifts them to automorphic forms having infinite product expansions and taking their zeros and poles along Heegner divisors. In order to transfer Borcherds-theory to unitary groups, we construct a suitable embedding of $\mathrm U (1,m)$ into $\mathrm O (2,2m)$ . We also derive a formula for the values taken by the Borcherds products at cusps of the symmetric domain of the unitary group. Further, as an application of the lifting, we obtain a modularity result for a generating series with Heegner divisors as coefficients, along the lines of Borcherds-generalization of the Gross-Zagier-Kohnen theorem.