Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Functional equations of $L$-functions for symmetric products of the Kloosterman sheaf
HTML articles powered by AMS MathViewer

by Lei Fu and Daqing Wan
Trans. Amer. Math. Soc. 362 (2010), 5947-5965
DOI: https://doi.org/10.1090/S0002-9947-2010-05172-4
Published electronically: June 14, 2010
PDF | Request permission

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
  • Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354652
  • H. Timothy Choi and Ronald Evans, Congruences for sums of powers of Kloosterman sums, Int. J. Number Theory 3 (2007), no. 1, 105–117. MR 2310495, DOI 10.1142/S1793042107000821
  • P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
  • R. J. Evans, Seventh power moments of Kloosterman sums, Israel J. Math., to appear.
  • R. J. Evans, Letter to N. Katz, Nov. 2005.
  • L. Fu, Calculation of $\ell$-adic local Fourier transformations, arXiv: 0702436 [math.AG] (2007).
  • Lei Fu and Daqing Wan, Trivial factors for $L$-functions of symmetric products of Kloosterman sheaves, Finite Fields Appl. 14 (2008), no. 2, 549–570. MR 2401995, DOI 10.1016/j.ffa.2007.07.005
  • Lei Fu and Daqing Wan, $L$-functions for symmetric products of Kloosterman sums, J. Reine Angew. Math. 589 (2005), 79–103. MR 2194679, DOI 10.1515/crll.2005.2005.589.79
  • Lei Fu and Daqing Wan, $L$-functions of symmetric products of the Kloosterman sheaf over $\textbf {Z}$, Math. Ann. 342 (2008), no. 2, 387–404. MR 2425148, DOI 10.1007/s00208-008-0240-5
  • K. Hulek, J. Spandaw, B. van Geemen, and D. van Straten, The modularity of the Barth-Nieto quintic and its relatives, Adv. Geom. 1 (2001), no. 3, 263–289. MR 1874236, DOI 10.1515/advg.2001.017
  • Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052, DOI 10.1515/9781400882120
  • G. Laumon, Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131–210 (French). MR 908218
  • Marko J. Moisio, The moments of a Kloosterman sum and the weight distribution of a Zetterberg-type binary cyclic code, IEEE Trans. Inform. Theory 53 (2007), no. 2, 843–847. MR 2302793, DOI 10.1109/TIT.2006.889020
  • Marko Moisio, On the moments of Kloosterman sums and fibre products of Kloosterman curves, Finite Fields Appl. 14 (2008), no. 2, 515–531. MR 2401992, DOI 10.1016/j.ffa.2007.06.001
  • 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
  • Philippe Robba, Symmetric powers of the $p$-adic Bessel equation, J. Reine Angew. Math. 366 (1986), 194–220. MR 833018, DOI 10.1515/crll.1986.366.194
  • Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
  • Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
  • Daqing Wan, Dwork’s conjecture on unit root zeta functions, Ann. of Math. (2) 150 (1999), no. 3, 867–927. MR 1740990, DOI 10.2307/121058
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
Bibliographic 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
  • MR Author ID: 195077
  • Email: dwan@math.uci.edu
  • 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).
  • © Copyright 2010 American Mathematical Society
  • 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
  • MathSciNet review: 2661503