## 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.