The limits of abelian reconstruction: twin curves exist, genus one suffices, and the obstruction is a torsor coordinate

Twin curves, non-isomorphic curves whose localized Bost-Connes systems are isomorphic as abelian systems, exist already in genus one: the point-count and Picard-class profiles are proved equal for isogenous elliptic curves (with the degree-n class equal to nu_n times a fixed class, nu_n = (phi^n - 1)/(phi - 1)), Lenstra's theorem supplies the group structures, and full enumerations over 11 <= p <= 29 verify every case. The reconstruction obstruction is exactly the torsor coordinate of Paper XIX.