arXiv subjects

Rufei Ren

Publications and source records attributed to Rufei Ren.

11 recordsLinked to original sources

arXiv2026

Coefficient-Level B\"ottcher Theory for Wild Superattracting Germs of Degree $p^e$

Let $p$ be an odd prime, let $e\ge2$, and put $q=p^e$. We study the wild family \[ \varphi_{r,e}(x)=x^q+qp^r x^{q+1}=x^{p^e}+p^{r+e}x^{p^e+1} \qquad (r\ge0), \] and the inverse B\"ottcher coordinate $f_{r,e}(x)=x\sum_{k\ge0}a_k(r,e)x^k/k!$ characterized by \[ \varphi_{r,e}(f_{r,e}(x))=f_{r,e}(x^q). \] For the clean family, we prove a complete mod-$p$ digit-sum law in the special fiber $r=0$. For the higher fibers $r\ge1$, we prove a coefficient-level theorem consisting of a global digit-weight lower bound, a leading monomial theorem on divisible non-pure classes, a lag-$e$ pure-power recursion, and subadditivity of the induced digit weight. This yields the pure-power branch word \[ (B^{e-1}A)^{\lceil r/e\rceil}B^\infty \] and the radius formula \[ \rho(f_{r,e})=p^{-\theta_{r,e}},\qquad \theta_{r,e}=p^{-e\lceil r/e\rceil}\left(\frac{1}{p-1}+e\lceil r/e\rceil-r\right). \] We then prove a tail-stable extension. In the special fiber, $p$-divisible tails preserve the digit-sum law modulo $p$. In the higher fibers, tails satisfying $v_p(\vartheta_h)\ge\Lambda_{r,e}(h+1)+1$ lie beyond the clean-family initial $\Lambda_{r,e}$-graded term and therefore preserve the leading terms, the pure-power branch word, the valuation asymptotic, and the radius. For $e=2$, this recovers the Salerno--Silverman degree-$p^2$ family and the Fu--Nie radius statement for the inverse coordinate in that family.

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arXiv2023

Non-Linearizability of power series over complete non-Archimedean fields of positive characteristic

In [H-Y83], Herman and Yoccoz prove that for any given locally analytic (at $z=0$) power series $f(z)=z(λ+\sum_{i=1}^\infty a_iz^i)$ over a complete non-Archimedean field of characteristic $0$ if $|λ|=1$ and $λ$ is not a root of unity, then $f$ is locally linearizable at $z=0$. They ask the same question for power series over fields 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. More precisely, given a complete non-Archimedean field $\mathcal K$ of positive characteristic and a power series $f(z)=z(λ+\sum_{i=1}^\infty a_iz^i) \in \mathcal K[\![z]\!]$ with $λ$ not a root of unity and $|1-λ|<1$, we prove a sufficient condition (Criterion~\star) for $f$ to be non-linearizable. This phenomenon of prevalence for power series over fields of positive characteristic being non-linearizable was initially conjectured in [Her87, p 147] by Herman, and formulated into a concrete question by Lindahl as [Lin04, Conjecture 2.2]. As applications of our criterion, we prove the non-linearizability of three families of polynomials.

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arXiv2022

The slope-invariant of local ghost series under direct sum

The ghost conjecture is first provided by Bergdall and Pollack in [BP-1,BP-2] to study the Up-slopes of spaces of modular forms, which, so far, has already brought plenty of important results. The local version of this conjecture under genericity condition has been solved by Liu-Truong-Xiao-Zhao in [LTXZ-1, LTXZ-2]. In the current paper, we prove a necessary and sufficient condition for a sequence of local ghost series to satisfy that their product has the same Newton polygon to the ghost series build from the direct sum of their associated modules. That answers a common question asked in both [BP2,LTXZ-1].

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arXiv2022

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.

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arXiv2021

Spectral halo for Hilbert modular forms

Let $F$ be a totally real field and $p$ be an odd prime which splits completely in $F$. We prove that the eigenvariety associated to a definite quaternion algebra over $F$ satisfies the following property: over a boundary annulus of the weight space, the eigenvariety is a disjoint union of countably infinitely many connected components which are finite over the weight space; on each fixed connected component, the ratios between the $U_\mathfrak{p}$-slopes of points and the $p$-adic valuations of the $\mathfrak{p}$-parameters are bounded by explicit numbers, for all primes $\mathfrak{p}$ of $F$ over $p$. Applying Hansen's $p$-adic interpolation theorem, we are able to transfer our results to Hilbert modular eigenvarieties. In particular, we prove that on every irreducible component of Hilbert modular eigenvarieties, as a point moves towards the boundary, its $U_p$ slope goes to zero. In the case of eigencurves, this completes the proof of Coleman-Mazur's `halo' conjecture.

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arXiv2020

Primitive prime divisors in the critical orbits of one-parameter families of rational polynomials

For a rational polynomial $f$ and rational numbers $c, u$, we put $f_c(x):=f(x)+c$, and consider the Zsigmondy set $\mathcal{Z}(f_c,u)$ associated to the sequence $\{f_c^n(u)-u\}_{n\geq 0}$, where $f_c^n$ is the $n$-st iteration of $f_c$. In this paper, we prove that if $u$ is a rational critical point of $f$, then there exists an $\mathbf M_f>0$ such that $\mathbf M_f\geq \max_{c\in \mathbb{Q}}\{\#\mathcal{Z}(f_c,u)\}$.

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arXiv2020

Iteration of Polynomials $AX^d+C$ Over Finite Fields

For a polynomial $f(X)=AX^d+C \in \mathbb{F}_p[X]$ with $A\neq 0$ and $d\geq 2$, we prove that if $d\;|\;p-1$ and $f^i(0)\neq f^j(0)$ for $0\leq i<j\leq N$, then $\#f^N(\mathbb{F}_p) \sim \frac{2p}{(d-1)N},$ where $f^N$ is the $N$-th iteration of $f$.

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arXiv2020

Generic Newton polygon for exponential sums in $n$ variables with parallelotope base

Let $p$ be a prime number. Every $n$-variable polynomial $f(\underline x)$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of varieties whose Galois group is isomorphic to $\mathbb{Z}_p$. Our goal of this paper is to study the Newton polygon of the $L$-function associated to a finite character of $\mathbb{Z}_p$ and a generic polynomial whose convex hull is an $n$-dimensional paralleltope $Δ$. We denote this polygon by $\mathrm{GNP}(Δ)$. We prove a lower bound of $\mathrm{GNP}(Δ)$, which is called the improved Hodge polygon $\mathrm{IHP}(Δ)$. We show that $\mathrm{IHP}(Δ)$ lies above the usual Hodge polygon $\mathrm{HP}(Δ)$ at certain infinitely many points, and when $p$ is larger than a fixed number determined by $Δ$, it coincides with $\mathrm{GNP}(Δ)$ at these points. As a corollary, we roughly determine the distribution of the slopes of $\mathrm{GNP}(Δ)$.

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arXiv2020

Newton slopes for twisted Artin--Schreier--Witt Towers

We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field of characteristic $p$ of degree relatively prime to $p$. Let $a\mapsto ω(a)$ be the Teichmüller lift of $\mathbb F_q$, and let $χ:\mathbb{Z}\to \mathbb C_p^\times$ be a finite character of $\mathbb Z_p$. The $L$-function associated to the polynomial $f$ and the so-called twisted character $ω^u\times χ$ is denoted by $L_f(ω^u,χ,s)$. We prove that, when the conductor of the character is large enough, the $p$-adic Newton slopes of this $L$-function form arithmetic progressions.

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arXiv2017

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 $\mathbb Z_p$. Our goal of this paper is to study the Newton polygon of the $L$-functions associated to a finite character of $\mathbb{Z}_p$ and a generic polynomial whose convex hull is a fixed triangle $Δ$. We denote this polygon by $\textrm{GNP}(Δ)$. We prove a lower bound of $\textrm{GNP}(Δ)$, which we call the improved Hodge polygon $\textrm{IHP}(Δ)$, and we conjecture that $\textrm{GNP}(Δ)$ and $\textrm{IHP}(Δ)$ are the same. We show that if $\textrm{GNP}(Δ)$ and $\textrm{IHP}(Δ)$ coincide at a certain point, then they coincide at infinitely many points. When $Δ$ 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 $\textrm{GNP}(Δ)$ and $\textrm{IHP}(Δ)$ coincide at infinitely many points. As a corollary, we deduce that the slopes of $\textrm{GNP}(Δ)$ roughly form an arithmetic progression with increasing multiplicities.

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arXiv2017

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 $\ell\geq 1$.

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