On the $L$-function of multiplicative character sums

Abstract

Let $\mathbb {F}_{q}$ be a finite field and let $\chi _{1}, \dots , \chi _{r}$ be multiplicative characters on $\mathbb {F}_{q}$. Suppose there are homogeneous polynomials $f_{1},\dots , f_{r}$ of degrees $d_{1}, \dots , d_{r}$ in $\mathbb {F}_{q} [x_{1}, \dots , x_{n}]$ and suppose that $f_{1}, \dots , f_{r}$ define smooth hypersurfaces in $\mathbb {P}^{n-1}$ that have normal crossings. When the character sum $S = \sum _{x \in \mathbb {P}^{n-1}(\mathbb {F}_{q})} \chi _{1}(f_{1}(x)) \dots \chi _{r}(f_{r}(x))$ is well defined, we compute the $p$-adic Dwork cohomology of the $L$-function associated to $S$. In particular, we give a lower bound for the $p$-adic Newton polygon of the $L$-function and give a formula for the Hilbert series of the non-vanishing cohomology in terms of $n,r,q$ and the $d_{j}$’s.