#### Vol. 15, No. 1, 2021

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$p$-adic estimates of exponential sums on curves

### Joe Kramer-Miller

Vol. 15 (2021), No. 1, 141–171
##### Abstract

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 $\stackrel{̄}{f}$ on $V$, we may form the $L$-function $L\left(\stackrel{̄}{f},V,s\right)$ associated to the exponential sums of $\stackrel{̄}{f}$. In this article, we prove a lower estimate on the Newton polygon of $L\left(\stackrel{̄}{f},V,s\right)$. 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 $ℤ∕pℤ$ in terms of local monodromy invariants.

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$p$-adic cohomology, Artin–Schreier covers, wild ramification, zeta function, Newton polygon, exponential sums