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arXiv · 1310.5311

Newton slopes for Artin-Schreier-Witt towers

Christopher Davis·Daqing Wan·Liang Xiao
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Abstract

We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field and consider the Artin-Schreier-Witt tower defined by $f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb A^1$, with total Galois group $\mathbb Z_p$. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function form arithmetic progressions which are independent of the conductor of the character. As a corollary, we obtain a result on the behavior of the slopes of the eigencurve associated to the Artin-Schreier-Witt tower, analogous to the result of Buzzard and Kilford.

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Christopher Davis, Daqing Wan, Liang Xiao. 2016-02-20. Newton slopes for Artin-Schreier-Witt towers. https://arxiv.org/abs/1310.5311

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