论文标题
$ p $ - 曲线上的Abelian Artin $ l $ functions的估计
$p$-adic estimates of abelian Artin $L$-functions on curves
论文作者
论文摘要
本文的目的是证明曲线上有限字符的“牛顿对霍奇”的结果。令$ x $为有限字段$ \ mathbb {f} _q $的特征性$ p \ geq 3 $，让$ v \ subset x $是仿射曲线的平滑曲线。考虑一个非平凡的有限字符$ρ：π_1^{et}（v）\ to \ mathbb {c}^\ times $。在本文中，我们证明了$ l $ function $ l（ρ，s）$的牛顿多边形的下限。估计值取决于$ρ$的单构不变的：天鹅指挥和当地指数。在某些非平稳化假设下，该下限与DeLigne引入的不规则霍奇过滤一致。特别是，我们的结果进一步表明了Deligne的预测，即不规则的Hodge过滤会迫使$ p $ - addic界限$ l $ - 功能。作为推论，我们可以在单层不变的方面获得对具有环状作用的曲线的牛顿多边形的估计。
The purpose of this article is to prove a "Newton over Hodge" result for finite characters on curves. Let $X$ be 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 $ρ:π_1^{et}(V) \to \mathbb{C}^\times$. In this article, we prove a lower bound on the Newton polygon of the $L$-function $L(ρ,s)$. The estimate depends on monodromy invariants of $ρ$: 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 force $p$-adic bounds on $L$-functions. As a corollary, we obtain estimates on the Newton polygon of a curve with a cyclic action in terms of monodromy invariants.
