The L-function of a non-degenerate twisted Witt extension is proved to be a polynomial. Its Newton polygon is proved to lie above the Hodge polygon of that extension. And the Newton polygons of the Gauss-Heilbronn sums are explicitly determined, generalizing the Stickelberger theorem.

In this article an upper bound for the first consecutive zeros of the Fermat quotient is given in terms of the zeros of a Mirimanoff polynomial. This bound is obtained by investigating a relation between these polynomials and the factor system of Gauss sums of a certain number field.

Abstract. In this paper, we construct two ternary linear codes C(SO(3, q)) and C(O(3, q)), respectively associated with the orthogonal groups SO(3, q) and O(3, q). Here q is a power of three. Then we obtain two recursive formulas for the power moments of Kloosterman sums with “trace nonzero square arguments” in terms of the frequencies of weights in the codes. This is done via Pless power momen...

In this paper, we construct the binary linear codes C(SL(n, q)) associated with finite special linear groups SL(n, q), with both n,q powers of two. Then, via Pless power moment identity and utilizing our previous result on the explicit expression of the Gauss sum for SL(n, q), we obtain a recursive formula for the power moments of multi-dimensional Kloosterman sums in terms of the frequencies o...

These notes are about applications of the one-dimensional double affine Hecke algebra to q-Gauss integrals and Gaussian sums. The double affine Hecke algebras were designed for a somewhat different purpose: to clarify the quantum Knizhnik-Zamolodchikov equation. Eventually (through the Macdonald polynomials) they led to a unification of the Harish-Chandra transform (the zonal case) and the p-ad...