Annihilators of the ideal class group of a cyclic extension of a global function field

Let $K$ be a global function field and fix a place $\infty$ of $K$. Let $L/K$ be a finite real abelian extension, i.e. a finite, abelian extension such that $\infty$ splits completely in $L$. Then we define a group of elliptic units $C_L$ in $\mathcal{O}_L^\times$ analogously to Sinnott's cyclotomic units and compute the index $[\mathcal{O}_L^\times:C_L]$. In the second part of this article, we additionally assume that $L$ is a cyclic extension of prime power degree. Then we can use the methods from Greither and Ku\v{c}era to take certain roots of these elliptic units and prove a result on the annihilation of the $p$-part of the class group of $L$.