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Exponents of Jacobians and relative class groups

Borys Kadets, Daniel Keliher

Abstract

We prove a lower bound for the exponent of the relative class group $\mathrm{Pic}^0 X_1/φ^* \mathrm{Pic}^0 X_2$ for a covering of curves $X_1 \to X_2$ over a finite field $\mathbb{F}_q$. The results improve on the existing best bounds (due to Stichtenoth) in the case $X_2=\mathbb{P}^1$, when the relative class group equals the class group of the function field $\mathbb{F}_q(X_1)$, and are completely new for the genuinely relative situation.

Exponents of Jacobians and relative class groups

Abstract

We prove a lower bound for the exponent of the relative class group for a covering of curves over a finite field . The results improve on the existing best bounds (due to Stichtenoth) in the case , when the relative class group equals the class group of the function field , and are completely new for the genuinely relative situation.

Paper Structure

This paper contains 5 sections, 6 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Outline and a sketch of the main argument
  3. Conventions
  4. Exponents of Jacobians
  5. Exponents of relative class groups

Key Result

Theorem 1.1

Suppose $X/\mathbb{F}_q$ is a smooth, proper, integral curve of genus $g$. Then the exponent $e$ of $\mathop{\mathrm{Pic}}\nolimits^0 X$ satisfies

Theorems & Definitions (12)

  • Theorem 1.1: Stichtenoth1979
  • Theorem 1.2
  • Theorem 1.3: Theorem \ref{['thm:relative-full']}
  • Remark 1.4
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 5.1
  • ...and 2 more