Evaluating Prime Power Gauss and Jacobi Sums

We show that for any mod $p^m$ characters, $\chi_1, \dots, \chi_k,$ the Jacobi sum, $$\sum_{x_1=1}^{p^m}\dots \sum_{\substack{x_k=1\\x_1+\dots+x_k=B}}^{p^m}\chi_1(x_1)\dots \chi_k(x_k),$$ has a simple evaluation when $m$ is sufficiently large (for $m\geq 2$ if $p\nmid B$). As part of the proof we give a simple evaluation of the mod $p^m$ Gauss sums when $m\geq 2$

Waring's number for large subgroups of double-struck Z_p

Let p be a prime, Z_p be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero k-th powers in Z_p. The goal of this paper is to determine, for a given positive integer s, a value t_s such that if |A| ≫ t_s then every element of Z_p is a sum of s k-th powers. We obtain t_4 = p^{\frac{22}{39} + \in}, t_5 = p^{\frac{15}{29} + \in} and for s s ≥ 6, t_s = p^{\frac{9s+45}{29s+33} + \in}. For s ≥ 24 further improvements are made, such as t_32 = p^{\frac{5}{16} + \in} and t_128 = p^{\frac{1}{4}}