Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

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

Author(s): Shaofang Hong
Journal: Proc. Amer. Math. Soc. 135 (2007), 3099-3108.
MSC (2000): Primary 11L03, 11T23, 14G10
Posted: June 20, 2007
MathSciNet review: 2322739
Retrieve article in: 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:

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: 10.1090/S0002-9939-07-08873-9
PII: S 0002-9939(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
Posted: 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
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia