Linear representation of endomorphisms of Kummer varieties

Let K A be a Kummer variety defined as the quotient of an Abelian variety A by the automorphism (−1) of A. Let T * 0 (A) be the co-tangent space at the point 0 of A. Let End(A) be the additive group of endomorphisms of A. There is a well defined map ρ : End(A) → Aut(T * 0 (A)), f → (df) * 0 , where (df) * 0 is the differential of f in 0 acting on T * 0 (A). The data of f ∈ End(K A) which comes from f ∈ End(A), determines ρ(f) up to a sign. The aim of this paper is to describe an efficient algorithm to recover ρ(f) up to a sign from the knowledge of f. Our algorithm is based on a study of the tangent cone of a Kummer variety in its singular 0 point. We give an application to Mestre's point counting algorithm.Cryptographie, isogenies et variété abéliennes surpuissante