Related papers: A classification of Newton polygons of L-functions…

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 $\chi$ when $p$ is larger than an explicit…

Let $\mathbb{F}_{q}$ denote the finite field of order $q$ (a power of a prime $p$). We study the $p$-adic valuations for zeros of $L$-functions associated with exponential sums of the following family of Laurent polynomials…

Let $\chi$ be an order $c$ multiplicative character of a finite field and $f(x)=x^d+\lambda x^e$ 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…

We study the $p$-adic absolute value of the roots of the $L$-functions associated to certain twisted character sums, and additive character sums associated to polynomials $P(x^d)$, when $P$ varies among the space of polynomial of fixed…

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible…

In this paper we consider the Newton polygons of $L$-functions coming from additive exponential sums associated to a polynomial over a finite field $\F_q$. These polygons define a stratification of the space of polynomials of fixed degree.…

Let p be a prime and let F_pbar be the algebraic closure of the finite field of p elements. Let f(x) be any one variable rational function over F_pbar with n poles of orders d_1, ...,d_n. Suppose p is coprime to d_i for every i. We prove…

The $L$-function of exponential sums associated to the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is studied. A polygon called the Frobenius polygon of the generic polynomial of degree $d$ in…

For a polynomial $f(x)$ in $(\mathbb{Z}_p\cap \mathbb{Q})[x]$ of degree $d>2$ let $L(f \bmod p;T)$ be the $L$-function of the exponential sum of $f \bmod p$. Let $\mathrm{NP}(f \bmod p)$ denote the Newton polygon of $L(f \bmod p;T)$. Let…

A conjecture of Le says that the Deligne polytope $\Delta_d$ is generically ordinary if $p\equiv 1\ (\!\!\bmod\ D(\Delta_d))$, where $D(\Delta_d)$ is a combinatorial constant determined by $\Delta_d$. In this paper a counterexample is given…

Let P(x) be a one-variable Laurent polynomial of degree (d_1,d_2) over a finite field of characteristic p. For any fixed positive integer s not divisible by p, we prove that the (normalized) p-adic Newton polygon of the L-functions of…

In 1984, Deligne proved that for any prime number $p$, the reduction modulo $p$ of the diagonal of a multivariate algebraic power series with integer coefficients is algebraic over the field of rational functions with coefficients in…

We study the p-adic valuations of roots of L-functions associated with certain families of exponential sums of Laurent polynomials in n variables over a finite field. The families we consider are reflection and Kloosterman variants of…

$T$-adic exponential sums associated to a Laurent polynomial $f$ are introduced. They interpolate all classical $p^m$-power order exponential sums associated to $f$. The Hodge bound for the Newton polygon of $L$-functions of $T$-adic…

The $p$-adic Newton polygon is a visual tool that encodes information about the roots and factorization of a polynomial relative to a prime $p$. In this article, we investigate how the Newton polygon changes under polynomial composition. If…

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,…

The $T$-adic exponential sum of a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the $C$-function of the…

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…

The $T$-adic exponential sum associated to a Laurent polynomial in one variable is studied. An explicit arithmetic polygon is proved to be the generic Newton polygon of the $C$-function of the T-adic exponential sum. It gives the generic…

This expository paper is based on the author's series of lectures delivered at the January 1999 Mini-course in Number Theory, held at Sogang University (Seoul). The aim is to give an elementary and self-contained introduction to the theory…