Gauss sums and Kloosterman sums over residue rings of algebraic integers

Abstract:

Let $\mathcal {O}$ denote the ring of integers of an algebraic number field of degree $m$ which is totally and tamely ramified at the prime $p$. Write $\zeta _q= \exp (2\pi i/q)$, where $q=p^r$. We evaluate the twisted Kloosterman sum \[ \sum \limits _{\alpha \in (\mathcal {O}/q \mathcal {O})^*} \chi (N(\alpha )) \zeta _q^{T(\alpha )+z/N(\alpha )},\] where $T$ and $N$ denote trace and norm, and where $\chi$ is a Dirichlet character (mod $q$). This extends results of Salié for $m=1$ and of Yangbo Ye for prime $m$ dividing $p-1.$ Our method is based upon our evaluation of the Gauss sum \begin{equation*}\sum \limits _{\alpha \in (\mathcal {O}/q\mathcal {O})^*} \chi (N(\alpha )) \zeta _q^{T(\alpha )},\end{equation*} which extends results of Mauclaire for $m=1$.