Slope Estimates of Artin-Schreier Curves

Jasper Scholten; Hui June Zhu

doi:10.1023/A:1024116216156

Slope Estimates of Artin-Schreier Curves

Jasper Scholten
, Hui June Zhu

Mathematics

KU Leuven

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

Let X/double struck F̄_p be an Artin-Schreier curve defined by the affine equation y^p - y = f̃(x) where f(x) εdouble struck F̄_p[x] is monic of degree d. In this paper we develop a method for estimating the first slope of the Newton polygon of X. Denote this first slope by NP₁(X/double struck F̄_p)). We use our method to prove that if p > d ≥ 2 then NP₁(X/double struck F̄_p) ≥ ⌈(p - 1)/d⌉/(p - 1). If p > 2d ≥ 4, we give a sufficient condition for the equality to hold.

Original language	English
Pages (from-to)	275-292
Number of pages	18
Journal	Compositio Mathematica
Volume	137
Issue number	3
DOIs	https://doi.org/10.1023/A:1024116216156
State	Published - Jul 2003

Keywords

Artin-Schreier curves
Exponential sums
Hodge polygon
Newton polygon
Zeta and L functions over finite fields

Access to Document

10.1023/A:1024116216156

Cite this

@article{4731f05243854ebda8ae99e045d9ec3d,

title = "Slope Estimates of Artin-Schreier Curves",

abstract = "Let X/double struck {\=F}p be an Artin-Schreier curve defined by the affine equation yp - y = {\~f}(x) where f(x) εdouble struck {\=F}p[x] is monic of degree d. In this paper we develop a method for estimating the first slope of the Newton polygon of X. Denote this first slope by NP1(X/double struck {\=F}p)). We use our method to prove that if p > d ≥ 2 then NP1(X/double struck {\=F}p) ≥ ⌈(p - 1)/d⌉/(p - 1). If p > 2d ≥ 4, we give a sufficient condition for the equality to hold.",

keywords = "Artin-Schreier curves, Exponential sums, Hodge polygon, Newton polygon, Zeta and L functions over finite fields",

author = "Jasper Scholten and Zhu, \{Hui June\}",

year = "2003",

month = jul,

doi = "10.1023/A:1024116216156",

language = "English",

volume = "137",

pages = "275--292",

journal = "Compositio Mathematica",

issn = "0010-437X",

publisher = "Cambridge University Press",

number = "3",

}

TY - JOUR

T1 - Slope Estimates of Artin-Schreier Curves

AU - Scholten, Jasper

AU - Zhu, Hui June

PY - 2003/7

Y1 - 2003/7

N2 - Let X/double struck F̄p be an Artin-Schreier curve defined by the affine equation yp - y = f̃(x) where f(x) εdouble struck F̄p[x] is monic of degree d. In this paper we develop a method for estimating the first slope of the Newton polygon of X. Denote this first slope by NP1(X/double struck F̄p)). We use our method to prove that if p > d ≥ 2 then NP1(X/double struck F̄p) ≥ ⌈(p - 1)/d⌉/(p - 1). If p > 2d ≥ 4, we give a sufficient condition for the equality to hold.

AB - Let X/double struck F̄p be an Artin-Schreier curve defined by the affine equation yp - y = f̃(x) where f(x) εdouble struck F̄p[x] is monic of degree d. In this paper we develop a method for estimating the first slope of the Newton polygon of X. Denote this first slope by NP1(X/double struck F̄p)). We use our method to prove that if p > d ≥ 2 then NP1(X/double struck F̄p) ≥ ⌈(p - 1)/d⌉/(p - 1). If p > 2d ≥ 4, we give a sufficient condition for the equality to hold.

KW - Artin-Schreier curves

KW - Exponential sums

KW - Hodge polygon

KW - Newton polygon

KW - Zeta and L functions over finite fields

UR - https://www.scopus.com/pages/publications/0347757103

U2 - 10.1023/A:1024116216156

DO - 10.1023/A:1024116216156

M3 - Article

AN - SCOPUS:0347757103

SN - 0010-437X

VL - 137

SP - 275

EP - 292

JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 3

ER -

Slope Estimates of Artin-Schreier Curves

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this