Abstract
AbstractThe purpose of this article is to prove a ‘Newton over Hodge’ result for finite characters on curves. LetXbe a smooth proper curve over a finite field$\mathbb {F}_q$of characteristic$p\geq 3$and let$V \subset X$be an affine curve. Consider a nontrivial finite character$\rho :\pi _1^{et}(V) \to \mathbb {C}^{\times }$. In this article, we prove a lower bound on the Newton polygon of theL-function$L(\rho ,s)$. The estimate depends on monodromy invariants of$\rho $: the Swan conductor and the local exponents. Under certain nondegeneracy assumptions, this lower bound agrees with the irregular Hodge filtration introduced by Deligne. In particular, our result further demonstrates Deligne’s prediction that the irregular Hodge filtration would forcep-adic bounds onL-functions. As a corollary, we obtain estimates on the Newton polygon of a curve with a cyclic action in terms of monodromy invariants.
Highlights
Let p be a prime with p ≥ 3 and let q = pa
The representation ρ is analogous to a rank 1 differential equation on a Riemann surface with regular singularities twisted by an exponential differential equation
We note that the work of Adolphson and Sperber treats the case of higher-dimensional tori as well
Summary
We know that the αi are l-adic units for any prime l ≠ p. We prove a ‘Newton over Hodge’ result This is in the vein of a celebrated theorem of Mazur [20], which compares the Newton and Hodge polygons of an algebraic variety over Fq. Our result differs from Mazur’s in that we study cohomology with coefficients in a local system. The representation ρ is analogous to a rank 1 differential equation on a Riemann surface with regular singularities twisted by an exponential differential equation (i.e., a weight 0 twisted Hodge module in the language of Esnault, Sabbah and Yu [9]). In this context one may define an irregular Hodge polygon [8, 9]. Our result gives further credence to the philosophy that characteristic 0 Hodge-type phenomena force p-adic bounds on lisse sheaves in characteristic p
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