Abelian Varieties with Prescribed Embedding Degree

Abstract

We present an algorithm that, on input of a CM-field $K$, an integer $k\ge 1$, and a prime $r\equiv 1\mathrm{mod}k$, constructs a $q$-Weil number $\pi \in {Ø}_K$ corresponding to an ordinary, simple abelian variety $A$ over the field $\F$ of $q$ elements that has an $\F$-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 $A$.

to appear in ANTS-VIII