# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## On the $L$-function of multiplicative character sumsHTML articles powered by AMS MathViewer

by John Dollarhide
Trans. Amer. Math. Soc. 365 (2013), 1637-1668 Request permission

## 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.
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