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Cubic-residue quadruplets: elliptic formulas and arithmetic fluctuations

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Brice Pouly ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

Let ApA_p count the monochromatic consecutive four-windows of a nontrivial cubic character modulo a prime p≡1(mod3)p\equiv1\pmod3, with zero excluded and overlapping windows counted separately. We give an exact formula for ApA_p in terms of fixed elliptic traces and a finite boundary correction. Its complex-multiplication terms are related cubic twists. This produces two arithmetic fluctuation regimes: the first four moments of (27Ap−p)/p(27A_p-p)/\sqrt p tend to (0,44,0,4224)(0,44,0,4224) when three is a cube and to (0,17,0,738)(0,17,0,738) otherwise. Over all admissible primes they tend to (0,26,0,1900)(0,26,0,1900). The proof combines a four-position identity following from Greene's hypergeometric formalism and Lennon's trace formula with symmetric-power automorphy, cyclic base change and a finite collection of Rankin-Selberg prime number theorems; it uses no GRH. A direct geometric proof exposes the phases and exceptional points, with the genus-two and isogeny constructions of Freeman-Satoh and Guillevic-Vergnaud explicitly credited. Point-order and Hasse certificates verify numerical examples without enumerating their billion-position character words. The conclusions concern fixed pattern length and four moments; no complete limiting distribution, constant-order centring of the raw count, or law for logarithmic-length maxima is proved.

Provenance statement

The following declaration describes the research and the published source edition: Artificial-intelligence tools assisted with proof development, computation, literature retrieval, and English writing and revision. Brice Pouly is the human author and assumes responsibility for the proof, programs, data, references and conclusions. Author-controlled checks do not constitute independent external replication or peer review. The accompanying article archive contains its numerical certificates and bounded reproduction commands. For this Hexagon deposit, OpenAI Codex assisted source packaging, compilation checks and the preparation of submission metadata. This preparation does not constitute independent mathematical peer review. The published edition used for this deposit, including its supplementary source material, is available at https://doi.org/10.5281/zenodo.23340847. At the time of preparation for Hexagon, the submitter declares the text primarily AI-generated and declares that no part of the contribution has been checked and understood by a human.

References

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Version history

  1. v1Submitted by Brice PoulyInitial depositCurrentOct 18, 2026