$L$-functions of twisted diagonal exponential sums over finite fields
Shaofang Hong · Proceedings of the American Mathematical Society · 2007
Let $\textbf {F}_q$ be the finite field of $q$ elements with characteristic $p$ and $\textbf {F}_{q^m}$ its extension of degree $m$. Fix a nontrivial additive character $\Psi$ and let $\chi _1,..., \chi _n$ be multiplicative characters of $\textbf {F}_p.$ For \[ f(x_1,...,x_n) \in \textbf {F}_q[x_1,x_1^{-1},...,x_n,x^{-1}_n],\] one can form the twisted exponential sum $S^*_m(\chi _1,...,\chi _n,f)$. The corresponding $L$-function is defined by \[ L^*(\chi _1,..., \chi _n,f;t)=\operatorname {exp}(\sum ^{\infty }_{m=0}S^*_m(\chi _1,...,\chi _n, f){\frac {t^m} {m}} ).\] In this paper, by using the $p$-adic gamma function and the Gross–Koblitz formula on Gauss sums, we give an explicit formula for the $L$-function $L^*(\chi _1,...,\chi _n, f;t)$ if $f$ is a Laurent diagonal polynomial. We also determine its $p$-adic Newton polygon.