Exponents of Jacobians and relative class groups

We prove a lower bound for the exponent of the relative class group ${\mathrm{P}\mathrm{i}\mathrm{c}}^0{X}_1\mathrm{/}{\phi }^{\ast }{\mathrm{P}\mathrm{i}\mathrm{c}}^0{X}_2$ 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.