Constructing genus 3 hyperelliptic Jacobians with CM

Jennifer Balakhrishnan (1) , Sorina Ionica (2) , Kristin Lauter (3) , Christelle Vincent (4)

1 Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom
2 MIS - Modélisation, Information et Systèmes - UR UPJV 4290
3 Microsoft Research [Redmond]
4 University of Vermont [Burlington]

Given a sextic CM field K, we give an explicit method for finding all genus-3 hyperelliptic curves defined over C whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339-372], we give an algorithm which works in complete generality, for any CM sextic field K, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field F-p, with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo p.

Domaines: Mathématique discrète [cs.DM]; Cryptographie et sécurité [cs.CR]

Jennifer Balakhrishnan, Sorina Ionica, Kristin Lauter, Christelle Vincent. Constructing genus 3 hyperelliptic Jacobians with CM. LMS Journal of Computation and Mathematics, 2016, 19 (Special Issue A: Algorithmic Number Theory Symposium XII), pp.283 - 300. ⟨10.1112/S1461157016000322⟩. ⟨hal-01323883⟩