The first slope case of Wan's conjecture

Jasper Scholten; Hui June Zhu

doi:10.1006/ffta.2001.0349

The first slope case of Wan's conjecture

Jasper Scholten
, Hui June Zhu

Mathematics

Radboud University Nijmegen

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

7 Scopus citations

Abstract

Let d ≥ 2 and p a prime coprime to d. For f(x) ε (ℤ_p ∩ ℚ)[x], let NP₁ (f mod p) denote the first slope of the Newton polygon of the L-function of the exponential sums We prove that there is a Zariski dense open subset U in the space A^d of degree-d monic polynomials over ℚ such that for all f(x) U we have lim_p→∞ NP₁ (f mod p) = 1/d. This is a "first slope case" of a conjecture of Wan.

Original language	English
Pages (from-to)	414-419
Number of pages	6
Journal	Finite Fields and their Applications
Volume	8
Issue number	4
DOIs	https://doi.org/10.1006/ffta.2001.0349
State	Published - Oct 2002

Keywords

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

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10.1006/ffta.2001.0349

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N2 - Let d ≥ 2 and p a prime coprime to d. For f(x) ε (ℤp ∩ ℚ)[x], let NP1 (f mod p) denote the first slope of the Newton polygon of the L-function of the exponential sums We prove that there is a Zariski dense open subset U in the space Ad of degree-d monic polynomials over ℚ such that for all f(x) U we have limp→∞ NP1 (f mod p) = 1/d. This is a "first slope case" of a conjecture of Wan.

AB - Let d ≥ 2 and p a prime coprime to d. For f(x) ε (ℤp ∩ ℚ)[x], let NP1 (f mod p) denote the first slope of the Newton polygon of the L-function of the exponential sums We prove that there is a Zariski dense open subset U in the space Ad of degree-d monic polynomials over ℚ such that for all f(x) U we have limp→∞ NP1 (f mod p) = 1/d. This is a "first slope case" of a conjecture of Wan.

KW - Artin-Schreier curves

KW - Exponential sums

KW - Hodge polygon

KW - Newton polygon

KW - Wan's conjecture

KW - Zeta and L functions over finite fields

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DO - 10.1006/ffta.2001.0349

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VL - 8

SP - 414

EP - 419

JO - Finite Fields and their Applications

JF - Finite Fields and their Applications

IS - 4

ER -

The first slope case of Wan's conjecture

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