Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. M. Artin, A. Grothendieck, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas (SGA 4), Lecture Notes in Math. 269, 270, 305, Springer-Verlag (1972-1973). MR 0354652 (50:7130); MR 0354653 (50:7131); MR 0354654 (50:7132)
  • 2. H. T. Choi and R. J. Evans, Congruences for sums of powers of Kloosterman sums, Inter. J. Number Theory, 3(2007), 105-117. MR 2310495 (2008d:11090)
  • 3. P. Deligne, Applications de la Formule des Traces aux Sommes Trigonométriques, in Cohomologie Étale (SGA $ 4\frac{1}{2}$), 168-232, Lecture Notes in Math. 569, Springer-Verlag, 1977. MR 0463174 (57:3132).
  • 4. R. J. Evans, Seventh power moments of Kloosterman sums, Israel J. Math., to appear.
  • 5. R. J. Evans, Letter to N. Katz, Nov. 2005.
  • 6. L. Fu, Calculation of $ \ell$-adic local Fourier transformations, arXiv: 0702436 [math.AG] (2007).
  • 7. L. Fu and D. Wan, Trivial factors for $ L$-functions of symmetric products of Kloosterman sheaves, Finite Fields and Appl. 14 (2008), 549-570. MR 2401995 (2009c:11135)
  • 8. L. Fu and D. Wan, $ L$-functions for symmetric products of Kloosterman sums, J. Reine Angew. Math. 589 (2005), 79-103. MR 2194679 (2006k:11159)
  • 9. L. Fu and D. Wan, $ L$-functions of symmetric products of the Kloosterman sheaf over $ {\mathbb{Z}}$, Math. Ann. 342 (2008), 387-404. MR 2425148 (2009i:14022)
  • 10. K. Hulek, J. Spandaw, B. van Geemen and D. van Straten, The modularity of the Barth-Nieto quintic and its relative, Adv. Geom., 1(2001), no.3, 263-289. MR 1874236 (2003b:11058)
  • 11. N. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Princeton University Press, 1988. MR 955052 (91a:11028)
  • 12. G. Laumon, Transformation de Fourier, constantes d'équations fonctionnelles, et conjecture de Weil, Publ. Math. IHES 65 (1987), 131-210. MR 908218 (88g:14019)
  • 13. M. Moisio, The moments of Kloosterman sums and the weight distribution of a Zetterberg-type binary cyclic code, IEEE, Trans. Inform. Theory, 53(2007), 843-847. MR 2302793 (2008c:11111)
  • 14. M. Moisio, On the moments of Kloosterman sums and fibre products of Kloosterman curves, Finite Fields Appl., 14(2008), 515-531. MR 2401992 (2009e:11226)
  • 15. C. Peters, J. Top and M. van der Vlugt, The Hasse zeta function of a K3 surface related to the number of words of weight $ 5$ in the Melas codes, J. Reine Angew. Math. 432(1992), 151-176. MR 1184764 (94d:11044)
  • 16. P. Robba, Symmetric powers of $ p$-adic Bessel functions, J. Reine Angew. Math., 366(1986), 194-220. MR 833018 (87f:12016)
  • 17. J.-P. Serre, Linear representations of finite groups, Springer-Verlag, 1977. MR 0450380 (56:8675)
  • 18. J.-P. Serre, Local fields, Springer-Verlag, 1979. MR 554237 (82e:12016)
  • 19. D. Wan, Dwork's conjecture on unit root zeta functions, Ann. of Math. (2), 150(1999), 867-927. MR 1740990 (2001a:11108)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11L05, 14G15

Retrieve articles in all journals with MSC (2000): 11L05, 14G15


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

American Mathematical Society