Non-Linearizability of power series over an untrametric field of positive characteristic

In [HY83], Herman and Yoccoz prove that every power series $f(T)=T(\lambda +\sum_{i=1}^\infty a_iT^i) \in \mathbb{Q}_p[\![T]\!]$ such that $|\lambda|=1$ and $\lambda$ is not a root of unity is linearizable. They asked the same question for power series in $\mathcal K[\![T]\!]$, where $\mathcal K$ is an ultrametric field of positive characteristic. In this paper, we prove that, on opposite, most such power series in this case are more likely to be non-linearizable, which was conjectured in [p 147, Her87] by Herman. More precisely, for $f(T)=T(\lambda +\sum\limits_{i=1}^\infty a_iT^i) \in \mathcal K[\![T]\!]$ such that $\lambda$ is not a root of unity and $0<|1-\lambda|<1$, we prove a sufficient condition (Criterion \star) of $f$ to be non-linearizable. By this criterion, we are able to show that a family of cubic polynomials over $\mathcal K$ is non-linearizable

Localized Gouv\^ea-Mazur conjecture

Gouv\^ea-Mazur [GM] made a conjecture on the local constancy of slopes of modular forms when the weight varies $p$-adically. Since one may decompose the space of modular forms according to associated residual Galois representations, the Gouv\^ea-Mazur conjecture makes sense for each such component. We prove the localized Gouv\^ea-Mazur conjecture when the residual Galois representation is irreducible and its restriction to $\textrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ is reducible and very generic

Slopes for higher rank Artin-Schreier-Witt Towers

We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$ over a finite field of characteristic $p$, and consider the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower defined by $\bar f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb{A}^1$, whose Galois group is canonically isomorphic to $\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified extension of $\mathbb{Z}_p$, which is abstractly isomorphic to $(\mathbb{Z}_p)^\ell$ as a topological group. 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 asymptotically form a finite union of arithmetic progressions. As a corollary, we prove the spectral halo property of the spectral variety associated to the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower. This extends the main result in [DWX] from rank one case $\ell=1$ to the higher rank case $\ell\geq 1$.Comment: 20 page

Generic Newton polygon for exponential sums in two variables with triangular base

Let $p$ be a prime number. Every two-variable polynomial $f(x_1, x_2)$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of surfaces whose Galois group is isomorphic to $\mathscr{Z}_p$. Our goal of this paper is to study the Newton polygon of the $L$-functions associated to a finite character of $\mathscr{Z}_p$ and a generic polynomial whose convex hull is a fixed triangle $\Delta$. We denote this polygon by \GNP(\Delta). We prove a lower bound of \GNP(\Delta), which we call the improved Hodge polygon \IHP(\Delta), and we conjecture that \GNP(\Delta) and \IHP(\Delta) are the same. We show that if \GNP(\Delta) and \IHP(\Delta) coincide at a certain point, then they coincide at infinitely many points. When $\Delta$ is an isosceles right triangle with vertices $(0,0)$, $(0, d)$ and $(d, 0)$ such that $d$ is not divisible by $p$ and that the residue of $p$ modulo $d$ is small relative to $d$, we prove that \GNP(\Delta) and \IHP(\Delta) coincide at infinitely many points. As a corollary, we deduce that the slopes of \GNP(\Delta) roughly form an arithmetic progression with increasing multiplicities