$p$-adic estimates of exponential sums on curves

DOI10.2140/ANT.2021.15.141zbMATH Open1470.11172arXiv1909.06905OpenAlexW2972985802MaRDI QIDQ2656319

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Publication date: 11 March 2021

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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 $m a t h b b F_{q}$ of characteristic $p g e q 3$ and let $V s u b s e t X$ be an affine curve. For a regular function $o v e r l i n e f$ on $V$ , we may form the $L$ -function $L (o v e r l i n e f, V, s)$ associated to the exponential sums of $o v e r l i n e f$ . In this article, we prove a lower estimate on the Newton polygon of $L (o v e r l i n e f, V, s)$ . The estimate depends on the local monodromy of $f$ around each point $x i n X - V$ . 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 $m a t h b b Z / p m a t h b b Z$ in terms of local monodromy invariants.

Full work available at URL: https://arxiv.org/abs/1909.06905

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\(p\)-adic estimates of exponential sums on curves