p-adic estimates of exponential sums on curves

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