The purpose of this article is to prove a “Newton over Hodge” result for exponential sums on curves. Let be a smooth proper curve over a finite field of characteristic and let be an affine curve. For a regular function on , we may form the -function associated to the exponential sums of . In this article, we prove a lower estimate on the Newton polygon of . The estimate depends on the local monodromy of around each point . This confirms a hope of Deligne that the irregular Hodge filtration forces bounds on -adic valuations of Frobenius eigenvalues. As a corollary, we obtain a lower estimate on the Newton polygon of a curve with an action of in terms of local monodromy invariants.
"$p$-adic estimates of exponential sums on curves." Algebra Number Theory 15 (1) 141 - 171, 2021. https://doi.org/10.2140/ant.2021.15.141