An inverse Jacobian algorithm for Picard curves

TL;DR

This paper presents a corrected algorithm for reconstructing Picard curves from period matrices, enabling the classification of CM Picard curves over complex fields and over rationals, with implications for algebraic geometry and number theory.

Contribution

It corrects a previous formula for the inverse Jacobian problem of Picard curves and applies this to classify CM Picard curves over complex and rational fields.

Findings

Corrected the inverse Jacobian algorithm for Picard curves.

Computed all isomorphism classes of CM Picard curves with certain properties.

Provided a conjectural complete list of CM Picard curves over .

Abstract

We study the inverse Jacobian problem for the case of Picard curves over $\mathbb{C}$. More precisely, we elaborate on an algorithm that, given a small period matrix $\Omega\in \mathbb{C}^{3\times 3}$ 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 $\mathbb{Q}$. 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].