Conference "Arithmetic, Geometry and Coding Theory"
March 14 - 18, 2011
CIRM, Marseille, France
Organisers: Yves Aubry (Université de la Méditerranée, IML), Christophe Ritzenthaler (Université de la Méditerranée, IML), Alexey Zykin (Mathematical Department of HSE, Laboratoire Poncelet, IITP)
Recent work of Lubicz and Robert (implemented in the AVIsogenies software package by Bisson, Cosset, and Robert) has given us an effective means of computing (l; l)-isogenies of abelian surfaces over finite fields. Their approach is based on theta functions, and requires a (possibly large) field extension to ensure enough theta 4l-level structure is rational. In this talk we describe an algebraic construction, avoiding (explicit) theta functions, which allows us to rapidly compute (3; 3)-isogenies with at most a quadratic field extension, avoiding the bottleneck introduced by the theta structure field extensions.