An inverse Jacobian algorithm for Picard curves
Résumé
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∈ C3 × 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 (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most 4. In particular, we obtain the complete list of maximal 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]. © 2021, The Author(s).
Domaines
Mathématiques [math]
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