Exponents of Jacobians and relative class groups

TL;DR

This paper establishes new lower bounds for the exponent of relative class groups in coverings of algebraic curves over finite fields, improving previous bounds and providing novel results for the relative case.

Contribution

It introduces improved lower bounds for the exponent of relative class groups in curve coverings, advancing understanding in algebraic geometry over finite fields.

Findings

Improved lower bounds for the exponent of relative class groups.

Results are new for genuinely relative situations.

Enhancements over previous bounds by Stichtenoth.

Abstract

We prove a lower bound for the exponent of the relative class group $\mathrm{Pic}^0 X_1/\phi^* \mathrm{Pic}^0 X_2$ 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.