The triple (C, c, m) is recoverable up to automorphism of the algebraic closure from the abstract group J_m(k-bar) and the embedded subset (C minus Supp(m))(k-bar).
Zilber’s Trichotomy in Hausdorff Geometric Structures.arXiv e-prints, page arXiv:2405.02209, May 2024
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A curve and its abstract generalized Jacobian
The triple (C, c, m) is recoverable up to automorphism of the algebraic closure from the abstract group J_m(k-bar) and the embedded subset (C minus Supp(m))(k-bar).