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Article Dans Une Revue Research in Number Theory Année : 2021

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).
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hal-03285430 , version 1 (13-07-2021)

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J.-C. Lario, A. Somoza, C. Vincent. An inverse Jacobian algorithm for Picard curves. Research in Number Theory, 2021, 7 (2), pp.32. ⟨10.1007/s40993-021-00253-1⟩. ⟨hal-03285430⟩
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