Computing class polynomials for abelian surfaces

Andreas Enge 1, 2 Emmanuel Thomé 3
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
3 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : We describe a quasi-linear algorithm for computing Igusa class polynomials of Jacobians of genus 2 curves via complex floating-point approximations of their roots. After providing an explicit treatment of the computations in quartic CM fields and their Galois closures, we pursue an approach due to Dupont for evaluating ϑ- constants in quasi-linear time using Newton iterations on the Borchardt mean. We report on experiments with our implementation and present an example with class number 20016.
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Andreas Enge, Emmanuel Thomé. Computing class polynomials for abelian surfaces. Experimental Mathematics, Taylor & Francis, 2014, 23, pp.129-145. ⟨10.1080/10586458.2013.878675⟩. ⟨hal-00823745v2⟩

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