Let p be a prime number. In this paper we use an old technique of Ore, based on Newton polygons, to construct in an efficient way p-integral bases of number fields defined by a p-regular equation. To illustrate the potential applications of this construction, we show how this result yields a computation of a p-integral basis of an arbitrary quartic field in terms of a defining equation.
In this paper, we study the Newton polygons for the $L$-functions of $n$-variable generalized Kloosterman sums. Generally, the Newton polygon has a topological lower bound, called the Hodge polygon. In order to determine the Hodge polygon, we explicitly construct a basis of the top dimensional Dwork cohomology. Using Wan's decomposition theorem and diagonal local theory, we obtain when the Newton polygon coincides with the Hodge polygon. In particular, we concretely get the slope sequence for $L$-function of $\bar{F}(\barλ,x):=\sum_{i=1}^nx_i^{a_i}+\barλ\prod_{i=1}^nx_i^{-1}$.
In this paper, we enumerate Newton polygons asymptotically. The number of Newton polygons is computable by a simple recurrence equation, but unexpectedly the asymptotic formula of its logarithm contains growing oscillatory terms. As the terms come from non-trivial zeros of the Riemann zeta function, an estimation of the amplitude of the oscillating part is equivalent to the Riemann hypothesis.
We review the Shimura-Taniyama method for computing the Newton polygon of an abelian variety with complex multiplication. We apply this method to cyclic covers of the projective line branched at three points. As an application, we produce multiple new examples of Newton polygons that occur for Jacobians of smooth curves in characteristic $p$. Under certain congruence conditions on $p$, these include: the supersingular Newton polygon for each genus $g$ with $4 \leq g \leq 11$; nine non-supersingular Newton polygons with $p$-rank $0$ with $4 \leq g \leq 11$; and, for all $g \geq 5$, the Newton p…
We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain PEL-type Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $p$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic cove…
We show that for a monic polynomial $f(x)$ over a number field $K$ containing a global permutation polynomial of degree $>1$ as its composition factor, the Newton Polygon of $f\mod\mathfrak p$ does not converge for $\mathfrak p$ passing through all finite places of $K$. In the rational number field case, our result is the "only if" part of a conjecture of Wan about limiting Newton polygons.
By a result of Moonen, there are exactly 20 positive-dimensional families of cyclic covers of the projective line for which the Torelli image is open and dense in the associated Shimura variety. For each of these, we compute the Newton polygons, and the $μ$-ordinary Ekedahl--Oort type, occurring in the characteristic $p$ reduction of the Shimura variety. We prove that all but a few of the Newton polygons appear on the open Torelli locus. As an application, we produce multiple new examples of Newton polygons and Ekedahl--Oort types of Jacobians of smooth curves in characteristic $p$. Under cert…
The purpose of this article is to study Newton polygons of certain abelian $L$-functions on curves. Let $X$ be a smooth affine curve over a finite field $\mathbb{F}_q$ and let $ρ:π_1(X) \to \mathbb{C}_p^\times$ be a finite character of order $p^n$. By previous work of the first author, the Newton polygon $\mathrm{NP}(ρ)$ lies above a `Hodge polygon' $\mathrm{HP}(ρ)$, which is defined using local ramification invariants of $ρ$. In this article we study the touching between these two polygons. We prove that $\mathrm{NP}(ρ)$ and $\mathrm{HP}(ρ)$ share a vertex if and only if a corresponding verte…
To understand L-function is an important fundamental question in Number Theory, but there are few specific results on it, especially the calculation of its Newton polygon. Following Dwork's method it is hard to calculate an exact example, even on the case of one variable. There are only three such examples till now, one of which has some mistakes. In this paper we calculate L-functions with p-adic Gauss sums and give a formula in power series(theorem 1.2.). After that we discuss Newton polygons NP(f/F_p,T) of L-functions of one variable polynomials and give a method to calculate its small slop…
In this article we study the behavior of Newton polygons along $\mathbb{Z}_p$-towers of curves. Fix an ordinary curve $X$ over a finite field $\mathbb{F}_q$ of characteristic $p$. By a $\mathbb{Z}_p$-tower $X_\infty/X$ we mean a tower of covers $ \dots \to X_2 \to X_1 \to X$ with $\mathrm{Gal}(X_n/X) \cong \mathbb{Z}/p^n\mathbb{Z}$. We show that if the ramification along the tower is sufficiently moderate, then the slopes of the Newton polygon of $X_n$ are equidistributed in the interval $[0,1]$ as $n$ tends to $\infty$. Under a stronger congruence assumption on the ramification invariants, we…
Let $χ$ be an order $c$ multiplicative character of a finite field and $f(x)=x^d+λx^e$ a binomial with $(d,e)=1$. We study the twisted classical and $T$-adic Newton polygons of $f$. When $p>(d-e)(2d-1)$, we give a lower bound of Newton polygons and show that they coincide if $p$ does not divide a certain integral constant depending on $p\bmod {cd}$. We conjecture that this condition holds if $p$ is large enough with respect to $c,d$ by combining all known results and the conjecture given by Zhang-Niu. As an example, we show that it holds for $e=d-1$.
A conjecture of Le says that the Deligne polytope $Δ_d$ is generically ordinary if $p\equiv 1\ (\!\!\bmod\ D(Δ_d))$, where $D(Δ_d)$ is a combinatorial constant determined by $Δ_d$. In this paper a counterexample is given to show that the conjecture is not true in general.
For prime $p\equiv-1\bmod d$ and $q$ a power of $p$, we obtain the slopes of the $q$-adic Newton polygons of $L$-functions of $x^d+ax^{d-1}\in \mathbb{F}_q[x]$ with respect to finite characters $χ$ when $p$ is larger than an explicit bound depending only on $d$ and $\log_p q$. The main tools are Dwork's trace formula and Zhu's rigid transform theorem.
Let $p$ be an odd prime. What are the possible Newton polygons for a curve in characteristic $p$? Equivalently, which Newton strata intersect the Torelli locus in $\mathcal{A}_g$? In this note, we study the Newton polygons of certain curves with $\mathbb{Z}/p\mathbb{Z}$-actions. Many of these curves exhibit unlikely intersections between the Torelli locus and the Newton stratification in $\mathcal{A}_g$. Here is one example of particular interest: fix a genus $g$. We show that for any $k$ with $\frac{2g}{3}-\frac{2p(p-1)}{3}\geq 2k(p-1)$, there exists a curve of genus $g$ whose Newton polygon …
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 $\math…
Considering the L-function of exponential sums associated to a polynomial over a finite field F_q, Deligne proved that a reciprocal root's p-adic order is a rational number in the interval [0, 1]. Based on hypergeometric theory, in this paper we improve this result that there are only finitely many possible forms of Newton polygons for the L-function of degree d polynomials independent of p, when p is larger than a constant D^*(theorem 4.3), i.e., a reciprocal root's p-adic order has form (up-v)/(D^*(p-1)) in which u, v have finitely many possible values. Furthermore, when p>D^*, to determine …
In this paper, we consider the following $(A, B)$-polynomial $f$ over finite field: $$f(x_0,x_1,\cdots,x_n)=x_0^Ah(x_1,\cdots,x_n)+g(x_1,\cdots,x_n)+P_B(1/x_0),$$ where $h$ is a Deligne polynomial of degree $d$, $g$ is an arbitrary polynomial of degree $< dB/(A+B)$ and $P_B(y)$ is a one-variable polynomial of degree $\le B$. Let $Δ$ be the Newton polyhedron of $f$ at infinity. We show that $Δ$ is generically ordinary if $p\equiv 1 \mod D$, where $D$ is a constant only determined by $Δ$. In other words, we prove that the Adolphson--Sperber conjecture is true for $Δ$.
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 $\textrm{GNP}(Δ)$ …
Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field $k$ of characteristic $p>0$ and $B \subset X(k)$ is a finite, possibly empty, set of points. The Newton polygon of a degree $p$ Galois cover of $X$ with branch locus $B$ depends on the ramification invariants of the cover. When $X$ is ordinary, for every possible set of branch points and ramification invariants, we prove that there exists such a cover whose Newton polygon is minimal or close to minimal.
We develop a theory of multiplicities of roots for polynomials over hyperfields and use this to provide a unified and conceptual proof of both Descartes' rule of signs and Newton's "polygon rule".
We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its properties on affinoid domains of the affine line, we prove that its slopes give rise to continuous functions that factorize by the retraction through a locally finite subgraph of the curve.
There are open questions about which Newton polygons and Ekedahl-Oort types occur for Jacobians of smooth curves of genus $g$ in positive characteristic $p$. In this chapter, I survey the current state of knowledge about these questions. I include a new result, joint with Karemaker, which verifies, for each odd prime $p$, that there exist supersingular curves of genus $g$ defined over $\overline{\mathbb F}_p$ for infinitely many new values of $g$. I sketch a proof of Faber and Van der Geer's theorem about the geometry of the $p$-rank stratification of the moduli space of curves. The chapter en…
Hodge-Stickelberger polygons for L-functions of exponential sums of P(x^s)
设P（x）是特征为p的有限域上（d1，d2）次的一元Laurent多项式，对任何不能被p整除的固定正整数s，证明了P（x^s）的指数和的L函数的（正规化）p-adic Newton多边形有一个紧的下界，我们称之为Hodge-Stickelberger多项式，它只依赖于d1，d2，s和（p mod s）.这个Hodge-Stickelberger多边形是P（x）的指数和的L-函数的“Hodge多边形”和x^s的指数和的L-函数的“Newton多边形”（由经典Stickelberger理论给出）的加权卷积。我们证明了一个类似的Hodge-Stickelberger下界的多变量劳伦多项式以及。我们证明了这个Hodge-Stickelberger多边形是P（x^s）的一般Newton多边形的极限，在本文中给出了明确的意义.
L-functions of Exponential sums over one-dimensional affinoid: Newton over Hodge
设p是素数，F_pbar是p元有限域的代数闭包.设f（x）是F_pbar上任意一个单变量有理函数，其n阶极点为d_1，.，d_n。设对每一个i，p与di互质证明了存在一个只依赖于di的Hodge多边形，它是f（x）的指数和的L函数的Newton多边形的一个下界.此外，我们证明了这两个多边形重合，如果p=1 mod d_i，对于每个i=1，...，n.作为推论，得到了Artin-Schreier曲线的Newton多边形的一个紧下界.
In this paper we give a modular interpretation of the $k$-th symmetric power $L$-function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of $p=2$ and $k$ odd give the precise 2-adic Newton polygon. We also give a $p$-adic modular interpretation of Dwork's unit root $L$-function of the Kloosterman family, and give the precise 2-adic Newton polygon when $k$ is odd. In a previous paper, we gave an estimate for the $q$-adic Newton polygon of the symmetric power $L$-function of the Kloosterman family when $p \geq 5$. We discuss how this restriction on prim…
On the classification of indecomposable Ekedahl-Oort strata in unitary Shimura varieties, and related Newton polygons
Washington College（华盛顿学院）；Carleton College（卡尔顿学院）；Oklahoma State University（俄克拉荷马州立大学）；Brooklyn College, City University of New York（纽约市立大学布鲁克林学院）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特维格-德弗里斯数学研究所）；Colorado State University（科罗拉多州立大学）
