The purpose of this article is to prove a “Newton over Hodge” result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field ${\mathbb{𝔽}}_q$ of characteristic $p\ge 3$ and let $V\subset X$ be an affine curve. For a regular function $\bar{f}$ on $V$, we may form the $L$-function $L\left(\bar{f},V,s\right)$ associated to the exponential sums of $\bar{f}$. In this article, we prove a lower estimate on the Newton polygon of $L\left(\bar{f},V,s\right)$. The estimate depends on the local monodromy of $f$ around each point $x\in X-V$. This confirms a hope of Deligne that the irregular Hodge filtration forces bounds on $p$-adic valuations of Frobenius eigenvalues. As a corollary, we obtain a lower estimate on the Newton polygon of a curve with an action of $\mathbb{Z}∕p\mathbb{Z}$ in terms of local monodromy invariants.

本文的目的是证明曲线上指数和的“牛顿超越霍奇”结果。让$X$ 在有限域上是一条平滑的适当曲线 ${\mathbb{𝔽}}_q$ 特征 $p\ge 3$ 然后让 $\mathrm{伏特}\subset X$是仿射曲线。对于常规功能$\bar{F}$ 在 $\mathrm{伏特}$，我们可能会形成 $\mathrm{大号}$-功能 $\mathrm{大号}（\bar{F}，\mathrm{伏特}，s）$ 与...的指数和有关 $\bar{F}$。在本文中，我们证明对$\mathrm{大号}（\bar{F}，\mathrm{伏特}，s）$。估计取决于当地的垄断$F$ 在每个点周围 $X\in X-\mathrm{伏特}$。这证实了Deligne的希望，即不规则的Hodge过滤力会在$p$Frobenius特征值的adic估值。作为推论，我们获得了曲线的牛顿多边形的较低估计，其作用为$\mathbb{Z}∕p\mathbb{Z}$ 就局部单变量不变而言。