$L$-functions of twisted diagonal exponential sums over finite fields
HTML articles powered by AMS MathViewer
- by Shaofang Hong PDF
- Proc. Amer. Math. Soc. 135 (2007), 3099-3108 Request permission
Abstract:
Let $\textbf {F}_q$ be the finite field of $q$ elements with characteristic $p$ and $\textbf {F}_{q^m}$ its extension of degree $m$. Fix a nontrivial additive character $\Psi$ and let $\chi _1,..., \chi _n$ be multiplicative characters of $\textbf {F}_p.$ For \[ f(x_1,...,x_n) \in \textbf {F}_q[x_1,x_1^{-1},...,x_n,x^{-1}_n],\] one can form the twisted exponential sum $S^*_m(\chi _1,...,\chi _n,f)$. The corresponding $L$-function is defined by \[ L^*(\chi _1,..., \chi _n,f;t)=\operatorname {exp}(\sum ^{\infty }_{m=0}S^*_m(\chi _1,...,\chi _n, f){\frac {t^m} {m}} ).\] 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
- Alan Adolphson and Steven Sperber, Exponential sums and Newton polyhedra: cohomology and estimates, Ann. of Math. (2) 130 (1989), no. 2, 367–406. MR 1014928, DOI 10.2307/1971424
- Alan Adolphson and Steven Sperber, $p$-adic estimates for exponential sums, $p$-adic analysis (Trento, 1989) Lecture Notes in Math., vol. 1454, Springer, Berlin, 1990, pp. 11–22. MR 1094845, DOI 10.1007/BFb0091132
- Bernard Dwork, On the zeta function of a hypersurface, Inst. Hautes Études Sci. Publ. Math. 12 (1962), 5–68. MR 159823
- B. Dwork, Normalized period matrices. I. Plane curves, Ann. of Math. (2) 94 (1971), 337–388. MR 396579, DOI 10.2307/1970865
- B. Dwork, Bessel functions as $p$-adic functions of the argument, Duke Math. J. 41 (1974), 711–738. MR 387281
- Benedict H. Gross and Neal Koblitz, Gauss sums and the $p$-adic $\Gamma$-function, Ann. of Math. (2) 109 (1979), no. 3, 569–581. MR 534763, DOI 10.2307/1971226
- Shaofang Hong, Newton polygons of $L$ functions associated with exponential sums of polynomials of degree four over finite fields, Finite Fields Appl. 7 (2001), no. 1, 205–237. Dedicated to Professor Chao Ko on the occasion of his 90th birthday. MR 1803945, DOI 10.1006/ffta.2000.0287
- Shaofang Hong, Newton polygons for $L$-functions of exponential sums of polynomials of degree six over finite fields, J. Number Theory 97 (2002), no. 2, 368–396. MR 1942966, DOI 10.1016/S0022-314X(02)00006-9
- S. Hong, Newton polygons for $L$-functions of exponential sums of polynomials of degree five over finite fields, preprint.
- Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
- B. Mazur, Frobenius and the Hodge filtration, Bull. Amer. Math. Soc. 78 (1972), 653–667. MR 330169, DOI 10.1090/S0002-9904-1972-12976-8
- Alain M. Robert, A course in $p$-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000. MR 1760253, DOI 10.1007/978-1-4757-3254-2
- S. Sperber, Congruence properties of the hyper-Kloosterman sum, Compositio Math. 40 (1980), no. 1, 3–33. MR 558257
- Steven Sperber, Newton polygons for general hyper-Kloosterman sums, Astérisque 119-120 (1984), 7, 267–330 (English, with French summary). $p$-adic cohomology. MR 773095
- Steven Sperber, On the $p$-adic theory of exponential sums, Amer. J. Math. 108 (1986), no. 2, 255–296. MR 833359, DOI 10.2307/2374675
- Da Qing Wan, Newton polygons and congruence decompositions of $L$-functions over finite fields, $p$-adic methods in number theory and algebraic geometry, Contemp. Math., vol. 133, Amer. Math. Soc., Providence, RI, 1992, pp. 221–241. MR 1183982, DOI 10.1090/conm/133/1183982
- Da Qing Wan, Newton polygons of zeta functions and $L$ functions, Ann. of Math. (2) 137 (1993), no. 2, 249–293. MR 1207208, DOI 10.2307/2946539
- 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
- Daqing Wan, Higher rank case of Dwork’s conjecture, J. Amer. Math. Soc. 13 (2000), no. 4, 807–852. MR 1775738, DOI 10.1090/S0894-0347-00-00339-8
- Daqing Wan, Higher rank case of Dwork’s conjecture, J. Amer. Math. Soc. 13 (2000), no. 4, 807–852. MR 1775738, DOI 10.1090/S0894-0347-00-00339-8
- Daqing Wan, Variation of $p$-adic Newton polygons for $L$-functions of exponential sums, Asian J. Math. 8 (2004), no. 3, 427–471. MR 2129244
- Roger Yang, Newton polygons of $L$-functions of polynomials of the form $x^d+\lambda x$, Finite Fields Appl. 9 (2003), no. 1, 59–88. MR 1954784, DOI 10.1016/S1071-5797(02)00006-0
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
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2322739