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

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Kunal Tyagi ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

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.

Provenance statement

Original version prepared with Claude (Anthropic); revised manuscript, mathematical audit and companion checks prepared with OpenAI Codex under Kunal Tyagi's direction. The revision explicitly corrects the scope of the earlier paper. Classical results are proved in the manuscript, with exact integer diagnostics. No independent human expert review of the revision is claimed. Original publication artifacts are at https://github.com/x67ai/riemann-rh-program; all new source and checks accompany this version. Zenodo concept DOI 10.5281/zenodo.22171136.

Tools used

Anthropic
ClaudeVersion original August 2026 work
OpenAI
CodexVersion October 2026 revision and checks

References

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Version history

  1. v1Submitted by Kunal TyagiInitial depositCurrentOct 18, 2026