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Linear representation of endomorphisms of Kummer varieties

David Lubicz 1 Damien Robert 2
2 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : 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.
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Submitted on : Wednesday, April 21, 2021 - 2:24:13 PM
Last modification on : Saturday, December 4, 2021 - 3:44:02 AM
Long-term archiving on: : Thursday, July 22, 2021 - 6:45:55 PM


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  • HAL Id : hal-03204365, version 1


David Lubicz, Damien Robert. Linear representation of endomorphisms of Kummer varieties. 2021. ⟨hal-03204365⟩



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