Explicit CM-theory for level 2-structures on abelian surfaces

For a complex abelian surface $A$ with endomorphism ring isomorphic to the maximal order in a quartic CM field $K$, the Igusa invariants $j_1\left(A\right),j_2\left(A\right),j_3\left(A\right)$ generate an unramified abelian extension of the reflex field of $K$. In this paper we give an explicit geometric description of the Galois action of the class group of this reflex field on $j_1\left(A\right),j_2\left(A\right),j_3\left(A\right)$. Our description can be expressed by maps between various Siegel modular varieties, and we can explicitly compute the action for ideals of small norm. We use the Galois action to modify the CRT method for computing Igusa class polynomials, and our run time analysis shows that this yields a significant improvement. Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby implying that the ‘isogeny volcano’ algorithm to compute endomorphism rings of ordinary elliptic curves over finite fields does not have a straightforward generalization to computing endomorphism rings of abelian surfaces over finite fields.