Isogeny, zeta-equivalence, and the Picard torsor in function field Bost-Connes systems

For function field Bost-Connes systems the deep stratum of the boundary is a torsor under the Picard group, and under the full gauge action of Paper XIV the closure of the translation monoid in its homeomorphism group is a compact group acting simply transitively, recovering the group structure and, with it, the torsion subgroup, hence the Jacobian. Consequently zeta-equivalent (isogenous) curves are distinguished exactly up to this torsor coordinate; the elliptic cases are enumerated explicitly.