Newton slopes for Artin-Schreier-Witt towers

Abstract

We fix a monic polynomial f (x) ∈ F q [x] over a finite field and consider the Artin-Schreier-Witt tower defined by f (x); this is a tower of curves · · · → C m → C m−1 → · · · → C 0 = A 1 , with total Galois group 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.

The 3-adic eigencurve at the boundary of weight space

“…In a related setting, Davis, Wan and Xiao [6] have recently proven that the eigencurve associated to the Artin-Schreier-Witt tower breaks up as a disjoint union of pieces over the boundary of weight space, with slopes in arithmetic progression.…”

The stable property of Newton slopes for general Witt towers

“…Recently, R.Ren, D.Wan, L.Xiao and M.Yu [5] proved a slope stable property in unit root case for higher rank (i.e., the Z l p -towers for l > 1), which is a different generalization of the main result of [1]. So, it will be natural to give a conjecture that there is a slope stable property for higher rank in our polynomial case.…”

“…By Lemma 1, we only need to study the Newton polygon of C * (χ, s). In the proof of the main theorem in [1], there is an important trick transferring between Newton polygons of C * (χ, s) and C * (T, s). We explain this trick by the following lemma.…”

“…The method is analogous to [1]. First, we consider the equivalent characteristic functions built out of L-functions.…”

“…The new difficulty, comparing with [1], is, in our case, if we apply T -adic valuation to the T -adic characteristic function, the p-power factor in the coefficients will be ignored. Then, we fail to produce a sharp enough lower bound of the π χ -adic Newton polygon to touch the upper bound.…”