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Functional equations of $ L$-functions for symmetric products of the Kloosterman sheaf


Authors: Lei Fu and Daqing Wan
Journal: Trans. Amer. Math. Soc. 362 (2010), 5947-5965
MSC (2000): Primary 11L05, 14G15
DOI: https://doi.org/10.1090/S0002-9947-2010-05172-4
Published electronically: June 14, 2010
MathSciNet review: 2661503
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Abstract: We determine the (arithmetic) local monodromy at 0 and at $ \infty$ of the Kloosterman sheaf using local Fourier transformations and Laumon's stationary phase principle. We then calculate $ \epsilon$-factors for symmetric products of the Kloosterman sheaf. Using Laumon's product formula, we get functional equations of $ L$-functions for these symmetric products and prove a conjecture of Evans on signs of constants of functional equations.


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Additional Information

Lei Fu
Affiliation: Institute of Mathematics, Nankai University, Tianjin, People’s Republic of China
Email: leifu@nankai.edu.cn

Daqing Wan
Affiliation: Department of Mathematics, University of California, Irvine, California 92697
Email: dwan@math.uci.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05172-4
Keywords: Kloosterman sheaf, $\epsilon$-factor, $\ell$-adic Fourier transformation
Received by editor(s): January 4, 2009
Published electronically: June 14, 2010
Additional Notes: The research of the first author was supported by the NSFC (10525107).
Article copyright: © Copyright 2010 American Mathematical Society