Paper 2008/061

Abelian varieties with prescribed embedding degree

David Freeman, Peter Stevenhagen, and Marco Streng

Abstract

We present an algorithm that, on input of a CM-field $K$, an integer $k\ge 1$, and a prime $r\equiv 1modk$, constructs a $q$-Weil number $\pi \in {\text{O}}_K$ corresponding to an ordinary, simple abelian variety $A$ over the field $\text{F}$ of $q$ elements that has an $\text{F}$-rational point of order $r$ and embedding degree $k$ with respect to $r$. We then discuss how CM-methods over $K$ can be used to explicitly construct $A$.