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Character sums in finite fields. (German) Zbl 0561.12008
Let \(q\) be a prime power, \(N_ m: {\mathbb{F}}^{\times}_{q^ m}\to {\mathbb{F}}_ q^{\times}\) the norm map, \(\chi_ 1,...,\chi_ b\) characters of \({\mathbb{F}}_ q\); \(\chi_ i^{(m)}=\chi_ i\cdot N_ m\) are extended with \(\chi_ i^{(m)}(0)=0\). Let \(\bar g_ 1,...,\bar g_ b\in {\mathbb{F}}_ q[x_ 1,...,x_ n]\). The paper studies the character sums \(S^*_ m=\sum \prod^{b}_{i=1}\chi_ i^{(m)}(\bar g_ i(x))\) where the sum is taken over all \(x\in ({\mathbb{F}}^{\times}_{q^ m})^ n\), and the associated L-series \(L^*(t)=\exp (\sum^{\infty}_{m=1}S^*_ m t^ m/m)\). The Dwork- Reich theory produces a completely continuous operator \(\alpha\) on a certain \(p\)-adic Banach space, such that the Fredholm determinant det(1- t\(\alpha)\) is closely related to \(L^*(t)\). The main result of the paper is a lower bound for the Newton-polygon of the entire function \(\text{det}(1-t\alpha)\). This bound can be used to estimate the degree of the rational function \(L^*\) and it gives information on the unit roots of \(L^*\).
Reviewer: M.van der Put

MSC:
11T24 Other character sums and Gauss sums
11L10 Jacobsthal and Brewer sums; other complete character sums
12J27 Krasner-Tate algebras
14G15 Finite ground fields in algebraic geometry
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References:
[1] A. Adolphson : An index theorem for p-adic differential operators . Trans. Amer. Math. Soc. 216 (1976) 279-293. · Zbl 0297.47039 · doi:10.2307/1997699
[2] A. Adolphson : Appendix to: B. Dwork , Lectures on p-adic differential equations, Grundlehren der mathematischen Wissenschaften 253. New York, Heidelberg, Berlin: Springer-Verlag (1982). · Zbl 0502.12021
[3] A. Adolphson and S. Sperber : Exponential sums on the complement of a hypersurface . Amer. J. Math. 102 (1980) 461-487. · Zbl 0444.12016 · doi:10.2307/2374109
[4] E. Bombieri : On exponential sums in finite fields . Amer. J. Math. 88 (1966) 71-105. · Zbl 0171.41504 · doi:10.2307/2373048
[5] E. Bombieri : On exponential sums in finite fields, II . Inventiones Math. 47 (1978) 29-39. · Zbl 0396.14001 · doi:10.1007/BF01609477 · eudml:142567
[6] H. Davenport : On character sums in finite fields . Acta Math. 71 (1939) 99-121. · Zbl 0021.20202 · doi:10.1007/BF02547751
[7] P. Deligne and N. Katz : Groupes de Monodromie en Géométrie Algébrique , Lecture Notes in Math. No. 340. Berlin, Heidelberg, New York: Springer-Verlag (1973). · Zbl 0258.00005
[8] B. Dwork : On the rationality of the zeta function of an algebraic variety . Amer. J. Math. 82 (1960) 631-648. · Zbl 0173.48501 · doi:10.2307/2372974
[9] B. Dwork : On the zeta function of a hypersurface . Publ. Math. I.H.E.S. 12 (1962) 5-68. · Zbl 0173.48601 · doi:10.1007/BF02684275 · numdam:PMIHES_1962__12__5_0 · eudml:103828
[10] B. Dwork : On the zeta function of a hypersurface, II . Ann. Math. 80 (1964) 227-299. · Zbl 0173.48601 · doi:10.1007/BF02684275 · numdam:PMIHES_1962__12__5_0 · eudml:103828
[11] (a) B. Dwork : Lectures on p-adic differential equations, Grundlehren der Mathematischen Wissenschaften 253. New York, Heidelberg, Berlin: Springer-Verlag (1982). · Zbl 0502.12021
[12] H. Hasse : Theorie der relativ-zyklischen algebraischen Funktionenkörper, inbesondere bei endlichem Konstantenkörper . J. Reine u. Angew. math. 172 (1934) 37-54. · Zbl 0010.00501 · crelle:GDZPPN002173026 · eudml:149898
[13] L. Illusie : Théorie de Brauer et caractéristique d’Euler-Poincaré . Astérisque 82 -83 (1981) 161-172. · Zbl 0496.14013
[14] D. Reich : A p-adic fixed point formula . Amer. J. Math. 91 (1969) 835-850. · Zbl 0213.47502 · doi:10.2307/2373354
[15] J-P. Serre : Endomorphismes completement continus des espaces de Banach p-adiques . Publ. Math. I.H.E.S. 12 (1962) 69-85. · Zbl 0104.33601 · doi:10.1007/BF02684276 · numdam:PMIHES_1962__12__69_0 · eudml:103829
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