Exponents of Jacobians and relative class groups

Abstract

We prove a lower bound for the exponent of the relative class group ${\mathrm{Pic}}^0{X}_1/\ddot{I}{•}^{\ast }{\mathrm{Pic}}^0{X}_2$ 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.

v2: 10 pages, expanded section 3, updated Theorem 4.3, minor corrections, accepted for publication by Proceedings of the AMS