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

TL;DR

This paper constructs elliptic units in global function fields and demonstrates how they can be used to annihilate the p-part of the ideal class group in cyclic extensions of prime power degree, extending class group theory.

Contribution

It introduces a new group of elliptic units in function fields and applies existing methods to show their role in annihilating class groups in specific cyclic extensions.

Findings

Computed the index of elliptic units in the unit group

Established annihilation results for the p-part of class groups

Extended techniques from number fields to function fields

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 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 $L$.