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On a Fiber Conjecture of Wan

Matthew Schmidt

Abstract

For a prime $p$ and $p$-power $q$, let $f(x)\in\mathbb{F}_q[x]$ with $\textrm{deg}\ f$ coprime to $p$. As $λ$ varies in $\overline{\mathbb{F}_p^\times}$, Wan has conjectured that the $p$-adic Newton polygon of the corresponding Artin-Schreier curve given by $λf$ is constant. That is, \[ \textrm{NP}(f) = \textrm{NP}(λf). \] In this paper, we prove this conjecture when $λ\in\mathbb{F}_p^\times$ and provide a detailed counterexample showing it is false in general.

On a Fiber Conjecture of Wan

Abstract

For a prime and -power , let with coprime to . As varies in , Wan has conjectured that the -adic Newton polygon of the corresponding Artin-Schreier curve given by is constant. That is, In this paper, we prove this conjecture when and provide a detailed counterexample showing it is false in general.
Paper Structure (6 sections, 24 theorems, 58 equations, 2 figures)

This paper contains 6 sections, 24 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $(\lambda,\mu)$-invariant Polynomials
  3. The Case $\lambda\in {\mathbb{F}_p^\times}$
  4. A Counterexample
  5. The Case $\lambda = 1$
  6. The Case $\lambda = \xi+2$

Key Result

Theorem 1.2

For $f(x)\in\mathbb{F}_q[x]$ and $\lambda\in\mathbb{F}_p^\times$, the Newton polygon is independent of $\lambda$:

Figures (2)

  • Figure 1: $\mathop{\mathrm{NP}}\nolimits(f)$ for $f(x)=x^8+x^6+x^2$
  • Figure 2: $\mathop{\mathrm{NP}}\nolimits(\lambda f)$ for $f = x^8+x^6+x^2$, $\lambda = \xi+2$

Theorems & Definitions (44)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Lemma 3.1
  • ...and 34 more