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Abstract

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 $C_L$ of elliptic units 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 of Greither and Kučera to take certain roots of these elliptic units and prove a result on the annihilation of the $p$-part of the class group of $L$.