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For a complex abelian variety $A$ with endomorphism ring isomorphic to themaximal order in a quartic CM-field $K$, the Igusa invariants $j_1(A),j_2(A),j_3(A)$ generate an abelian extension of the reflex field of $K$. Inthis paper we give an explicit description of the Galois action of the classgroup of this reflex field on $j_1(A),j_2(A),j_3(A)$. We give a geometricdescription which can be expressed by maps between various Siegel modularvarieties. We can explicitly compute this action for ideals of small norm, andthis allows us to improve the CRT method for computing Igusa class polynomials.Furthermore, we find cycles in isogeny graphs for abelian surfaces, therebyimplying that the `isogeny volcano' algorithm to compute endomorphism rings ofordinary elliptic curves over finite fields does not have a straightforwardgeneralization to computing endomorphism rings of abelian surfaces over finitefields.

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