An inverse Jacobian algorithm for Picard curves

2016/11/08 by Lario, Joan-C., Somoza, Anna, Vincent, Christelle
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1611.02582

Abstract

We study the inverse Jacobian problem for the case of Picard curves over ℂ. More precisely, we elaborate on an algorithm that, given a small period matrix Ω∈ ℂ^{3× 3} corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre-Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike-Weng in [Math. Comp., 74(249):499-518, 2005] which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain (numerically) all the isomorphism classes of Picard curves with maximal complex multiplication attached to the sextic CM-fields with class number at most 4. In particular, we obtain (conjecturally) the complete list of CM Picard curves defined over ℚ. In the appendix, Vincent gives a correction to the generalization of Takase's formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan-Ionica-Lauter-Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016].

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