#### Vol. 5, No. 4, 2011

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Explicit CM theory for level 2-structures on abelian surfaces

### Reinier Bröker, David Gruenewald and Kristin Lauter

Vol. 5 (2011), No. 4, 495–528
##### Abstract

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.

##### Keywords
abelian surface, isogeny, level structure
Primary: 11G15

Magma code

##### Milestones
Received: 23 October 2009
Revised: 18 April 2011
Accepted: 12 July 2011
Published: 21 December 2011
##### Authors
 Reinier Bröker Department of Mathematics Brown University Box 1917 151 Thayer Street Providence, RI 02912 United States David Gruenewald Laboratoire de Mathématiques Nicolas Oresme CNRS UMR 5139, UFR Sciences Campus 2, Boulevard Maréchal Juin Université de Caen Basse-Normandie 14032 Caen cedex France Kristin Lauter Microsoft Research One Microsoft Way Redmond, WA 98052 United States