Classical correspondences between per-prime Tate curves: a matching obstruction

For the complex Tate curves E_p = C*/p^Z attached to primes, we compute the classical divisorial correspondence groups. For distinct primes, Hom(E_p,E_q) is nonzero exactly when log(p)log(q) belongs to 4 pi^2 Q. If log(p)log(q)/(4 pi^2) = u/v in lowest positive terms, this group has rank one and its generator has isogeny degree uv. The isogeny relation between distinct primes is a partial matching: unconditionally, each prime has at most one partner. These rigidity statements specialize results of Winkelmann; the explicit Hom generator and degree are included here. Outside the matching, the Neron-Severi group of E_p x E_q consists of the two fiber classes, which act trivially on H^1. On every diagonal surface End(E_p) = Z, by Gelfond-Schneider, and the intersection profile of multiplication graphs is independent of p. Consequently no connected classical correspondence system can involve three distinct prime fibers. Version 2 withdraws the broader universal no-go assertion of version 1 and states the exact cross-fiber support restriction required by the obstruction.