Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$ L$-functions of twisted diagonal exponential sums over finite fields


Author: Shaofang Hong
Journal: Proc. Amer. Math. Soc. 135 (2007), 3099-3108
MSC (2000): Primary 11L03, 11T23, 14G10
DOI: https://doi.org/10.1090/S0002-9939-07-08873-9
Published electronically: June 20, 2007
MathSciNet review: 2322739
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\bf F}_q$ be the finite field of $ q$ elements with characteristic $ p$ and $ {\bf F}_{q^m}$ its extension of degree $ m$. Fix a nontrivial additive character $ \Psi$ and let $ \chi _1,..., \chi _n$ be multiplicative characters of $ {\bf F}_p.$ For

$\displaystyle f(x_1,...,x_n) \in {\bf F}_q[x_1,x_1^{-1},...,x_n,x^{-1}_n],$

one can form the twisted exponential sum $ S^*_m(\chi _1,...,\chi _n,f)$. The corresponding $ L$-function is defined by

$\displaystyle L^*(\chi _1,..., \chi _n,f;t)=\hbox {exp}(\sum^{\infty}_{m=0}S^*_m(\chi _1,...,\chi _n, f){\frac {t^m} {m}} ).$

In this paper, by using the $ p$-adic gamma function and the Gross-Koblitz formula on Gauss sums, we give an explicit formula for the $ L$-function $ L^*(\chi _1,...,\chi _n, f;t)$ if $ f$ is a Laurent diagonal polynomial. We also determine its $ p$-adic Newton polygon.


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

  • 1. A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra: Cohomology and estimates, Ann. of Math.(2) 130 (1989), 367-406. MR 1014928 (91e:11094)
  • 2. A. Adolphson and S. Sperber, $ p$-Adic estimates for exponential sums, Lectures Notes in Math. 1454 (1990), 367-406. MR 1094845 (92d:11086)
  • 3. B. Dwork, On the zeta function of a hypersurface, Publ. Math. I.H.E.S. 12 (1962), 5-68. MR 0159823 (28:3039)
  • 4. B. Dwork, Normalized period matrices II, Ann. of Math.(2) 98 (1973), 1-57. MR 0396580 (53:442b)
  • 5. B. Dwork, Bessel functions as $ p$-adic functions of the argument, Duke Math. J. 41 (1974), 711-738. MR 0387281 (52:8124)
  • 6. B.H. Gross and N. Koblitz, Gauss sums and the $ p$-adic $ \Gamma$-function, Ann. of Math.(2) 109 (1979), 569-581. MR 534763 (80g:12015)
  • 7. S. Hong, Newton polygons of L-functions associated with exponential sums of polynomials of degree four over finite fields, Finite Fields Appl. 7 (2001), 205-237. MR 1803945 (2001j:11120)
  • 8. S. Hong, Newton polygons for $ L$-functions of exponential sums of polynomials of degree six over finite fields, J. Number Theory 97 (2002), 368-396. MR 1942966 (2003i:11116)
  • 9. S. Hong, Newton polygons for $ L$-functions of exponential sums of polynomials of degree five over finite fields, preprint.
  • 10. K. Ireland and M. Rosen, A classical introduction to modern number theory, GTM 84, Springer-Verlag, New York, 2nd edition, 1990. MR 1070716 (92e:11001)
  • 11. B. Mazur, Frobenius and the Hodge filtration, Bull. A.M.S. 78 (1972), 635-667. MR 0330169 (48:8507)
  • 12. A.M. Robert, A course in $ p$-adic analysis, GTM 198, Springer-Verlag, New York, 2000. MR 1760253 (2001g:11182)
  • 13. S. Sperber, Congruence properties of the hyper-Kloosterman sum, Compositio Math. 40 (1980), 3-33. MR 558257 (81j:10059)
  • 14. S. Sperber, Newton polygons for general hyper-Kloosterman sums, Astérisque 119-120 (1984), 267-330. MR 773095 (86i:11044)
  • 15. S. Sperber, On the $ p$-adic theory of exponential sums, Amer. J. Math. 108 (1986), 255-296. MR 833359 (87j:11055)
  • 16. D. Wan, Newton polygons and congruence decompositions of L-functions over finite fields, Contemp. Math. 133 (1992), 221-241. MR 1183982 (93i:11099)
  • 17. D. Wan, Newton polygons of zeta functions and $ L$-functions, Ann. of Math.(2) 137(1993), 249-293. MR 1207208 (94f:11074)
  • 18. D. Wan, Dwork's conjecture on unit root zeta functions, Ann. of Math.(2) 150 (1999), 867 -927. MR 1740990 (2001a:11108)
  • 19. D. Wan, Higher rank case of Dwork's conjecture, J. Amer. Math. Soc. 13 (2000), 807-852. MR 1775738 (2001f:11101a)
  • 20. D. Wan, Rank one case of Dwork's conjecture, J. Amer. Math. Soc. 13 (2000), 853-908. MR 1775761 (2001f:11101b)
  • 21. D. Wan, Variation of $ p$-adic Newton polygons for L-functions of exponential sums, Asian J. Math. 8 (2004), 427-472. MR 2129244 (2006b:11095)
  • 22. R. Yang, Newton polygons of $ L$-functions of polynomials of the form $ x^d + \lambda x$, Finite Fields Appl. 9 (2003), 59-88. MR 1954784 (2003i:11180)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11L03, 11T23, 14G10

Retrieve articles in all journals with MSC (2000): 11L03, 11T23, 14G10


Additional Information

Shaofang Hong
Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China
Email: s-f.hong@tom.com, hongsf02@yahoo.com

DOI: https://doi.org/10.1090/S0002-9939-07-08873-9
Keywords: Twisted exponential sum, $L$-function, $p$-adic Newton polygon, $\Gamma$-function, Gross--Koblitz formula, Hasse--Davenport relation.
Received by editor(s): May 1, 2006
Received by editor(s) in revised form: July 20, 2006
Published electronically: June 20, 2007
Additional Notes: The research of this author was supported by New Century Excellent Talents in University Grant # NCET-060785, by SRF for ROCS, SEM and by NNSF of China Grant # 10101015
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society