Polynomials for Kloosterman Sums

Gurak, S.

Canadian mathematical bulletin, Tome 50 (2007) no. 1, pp. 71-84

Résumé

Fix an integer $m>1$ , and set ${{\zeta }_{m}}=\exp \left( 2\pi i/m \right)$ . Let $\bar{x}$ denote the multiplicative inverse of $x$ modulo $m$ . The Kloosterman sums $R\left( d \right)=\sum\limits_{x}{\zeta _{m}^{x+d\bar{x}}},1\le d\le m,\left( d,m \right)=1$ , satisfy the polynomial $${{f}_{m}}\left( x \right)=\underset{d}{\mathop{\prod }}\,\left( x-R\left( d \right) \right)={{x}^{\phi \left( m \right)}}+{{c}_{1}}{{x}^{\phi \left( m \right)-1}}+\cdot \cdot \cdot +{{c}_{\phi \left( m \right)}},$$ where the sum and product are taken over a complete system of reduced residues modulo $m$ . Here we give a natural factorization of ${{f}_{m}}\left( x \right)$ , namely, $${{f}_{m}}\left( x \right)=\underset{\sigma }{\mathop{\prod }}\,f_{m}^{\left( \sigma\right)}\left( x \right),$$ where $\sigma$ runs through the square classes of the group $Z_{m}^{*}$ of reduced residues modulo $m$ . Questions concerning the explicit determination of the factors $f_{m}^{\left( \sigma\right)}\left( x \right)$ (or at least their beginning coefficients), their reducibility over the rational field $\text{Q}$ and duplication among the factors are studied. The treatment is similar to what has been done for period polynomials for finite fields.

Détail

DOI : 10.4153/CMB-2007-007-2

Mots-clés : 11L05, 11T24

Bibliographie

[1] [1] Berndt, B. C., Evans, R. J. and Williams, K. S., Gauss and Jacobi Sums. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley-Interscience, New York, 1998. Google Scholar

[2] [2] Borevich, Z. and Shafarevich, I., Number Theory. Pure and Applied Mathematics 20, Academic Press, New York, 1966. Google Scholar

[3] [3] Gaal, L., Classical Galois Theory with Examples. Chelsea, New York, 1973. Google Scholar

[4] [4] Gupta, S. and Zagier, D., On the coefficients of the minimal polynomial of Gaussian periods. Math. Comp. 60(1993), 385–398. Google Scholar

[5] [5] Gurak, S., Minimal polynomials for circular numbers. Pacific J. Math 112(1984), no. 2, 313–331. Google Scholar

[6] [6] Gurak, S., Factors of period polynomials for finite fields. I. In: The Rademacher Legacy toMathematics, Contemp. Math. 166, American Mathematical Society, Providence, RI, 1994), pp. 309–333. Google Scholar

[7] [7] Gurak, S., On the minimal polynomials for certain Gauss periods over finite fields. In: Finite Fields and their Applications, LondonMath. Soc. Lecture Note Ser. 233, Cambridge, Cambridge University Press, 1996, pp. 85–96. Google Scholar

[8] [8] Gurak, S., Minimal polynomials for Gauss periods with f = 2 . Acta Arith. 121(2006), 233–257. Google Scholar

[9] [9] Iwaniec, H., Topics in classical automorphic forms. Graduate Studies in Mathematics 17, American Mathematical Society, Providence, RI, 1997. Google Scholar

[10] [10] Lehmer, D., On the cubes of Kloosterman sums. Acta Arith. 6(1960), 15–22. Google Scholar

[11] [11] Myerson, G., Period polynomials and Gauss sums for finite fields. Acta Arith. 39(1981), no. 3, 251–264. Google Scholar

[12] [12] Salie, H., Über die Kloostermanschen Summen S(u, v; q) . Math Z. 34(1932), no. 1, 91–109. Google Scholar

[13] [13] Sarnak, P., Some applications of modular forms. Cambridge Tracts in Mathematics 99, Cambridge University Press, Cambridge, 1990. Google Scholar