Vol. 5, No. 4, 2011

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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 j1(A),j2(A),j3(A) 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 j1(A),j2(A),j3(A). 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
Mathematical Subject Classification 2000
Primary: 11G15
Supplementary material

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