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

TL;DR

This paper provides an explicit description of Galois actions on Igusa invariants for abelian surfaces with complex multiplication, improving computational methods and revealing limitations of existing algorithms for endomorphism ring calculations.

Contribution

It explicitly describes Galois actions on Igusa invariants via geometric maps, enhancing class polynomial computations and analyzing isogeny graphs for abelian surfaces.

Findings

Explicit Galois action formulas for Igusa invariants

Improved CRT method for Igusa class polynomial computation

Identification of cycles in isogeny graphs affecting endomorphism ring algorithms

Abstract

For a complex abelian variety $A$ with endomorphism ring isomorphic to the maximal 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 $K$. In this paper we give an explicit description of the Galois action of the class group of this reflex field on $j_1(A),j_2(A),j_3(A)$. We give a geometric description which can be expressed by maps between various Siegel modular varieties. We can explicitly compute this action for ideals of small norm, and this allows us to improve the CRT method for computing Igusa class polynomials. 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.