Hexagon

math.NT — Number Theory

Arithmetic structure in a complete atlas of power-residue runs below $2^{32}$ : Rabung profiles, Kummer densities, norm dependencies, and local codes

Contributed by Brice Pouly

We study the arithmetic behind long runs in a complete census of power-residue colourings below $2^{32}$ , covering every admissible order $2\le r\le9$ . The dataset contains $203\,280\,221$ primes and $677\,591\,249$ profiles, with exact quadratic four-window counts at every odd prime. A bounded study of its deep tail identifies algebraic windows with fewer independent character conditions than their lengths suggest. We certify their power-class groups and conjugate-root overlaps, obtaining exact Kummer densities for fixed templates, including eight additional order-four models. In a quadratic family, eleven constant positions force twenty-six neutral positions at odd character orders. In two further families, norm identities generate all remaining higher-root relations. A length- $53$ cubic realizes a three-wise independent binary code whose monochromatic probability is $4096$ times that of independent signs; its local run law is computed exactly. A separate stratified million-profile ternary sample is analysed with the same finite catalogue filter applied to the data and to an independent-colour comparator. Finite census statements, selected explanations and asymptotic template laws concern distinct populations. No universal law of global maxima or finite-range error estimate for the template densities is asserted.

Primarily AI-generated textHuman understanding: no partsKummer densitiesRabung profilesarithmetic dependenciescomplete finite atlaslocal codesmultiplicative charactersnorm identitiespower-residue runs

Oct 19, 2026PDF ↗View work →

hexagon:2610.00362v1

math.NT — Number Theory

Cubic-residue quadruplets: elliptic formulas and arithmetic fluctuations

Contributed by Brice Pouly

Let $A_p$ count the monochromatic consecutive four-windows of a nontrivial cubic character modulo a prime $p\equiv1\pmod3$ , with zero excluded and overlapping windows counted separately. We give an exact formula for $A_p$ 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 $(27A_p-p)/\sqrt p$ tend to $(0,44,0,4224)$ when three is a cube and to $(0,17,0,738)$ otherwise. Over all admissible primes they tend to $(0,26,0,1900)$ . 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.

Primarily AI-generated textHuman understanding: no partsElliptic Curvesarithmetic fluctuationsaveraged momentscomplex multiplicationcubic characterscubic twistsfinite fieldsfour-term patterns

Oct 18, 2026PDF ↗View work →

Matching works

Arithmetic structure in a complete atlas of power-residue runs below $2^{32}$ : Rabung profiles, Kummer densities, norm dependencies, and local codes

Cubic-residue quadruplets: elliptic formulas and arithmetic fluctuations

Advanced search

math.AC — Commutative Algebra
math.AG — Algebraic Geometry
math.AP — Analysis of PDEs
math.AT — Algebraic Topology
math.CA — Classical Analysis and ODEs
math.CO — Combinatorics
math.CT — Category Theory
math.CV — Complex Variables
math.DG — Differential Geometry
math.DS — Dynamical Systems
math.FA — Functional Analysis
math.GM — General Mathematics
math.GN — General Topology
math.GR — Group Theory
math.GT — Geometric Topology
math.HO — History and Overview
math.IT — Information Theory
math.KT — K-Theory and Homology
math.LO — Logic
math.MG — Metric Geometry
math.NA — Numerical Analysis
math.NT — Number Theory
math.OA — Operator Algebras
math.OC — Optimization and Control
math.PR — Probability
math.QA — Quantum Algebra
math.RA — Rings and Algebras
math.RT — Representation Theory
math.SG — Symplectic Geometry
math.SP — Spectral Theory
math.ST — Statistics Theory
cs.AI — Artificial Intelligence
cs.CC — Computational Complexity
cs.CG — Computational Geometry
cs.DM — Discrete Mathematics
cs.DS — Data Structures and Algorithms
cs.FL — Formal Languages and Automata Theory
cs.IT — Information Theory
cs.LO — Logic in Computer Science
cs.MS — Mathematical Software
cs.NA — Numerical Analysis
cs.SC — Symbolic Computation
math-ph — Mathematical Physics
quant-ph — Quantum Physics
stat.TH — Statistics Theory

Matching works

Arithmetic structure in a complete atlas of power-residue runs below 2322^{32}232: Rabung profiles, Kummer densities, norm dependencies, and local codes

Cubic-residue quadruplets: elliptic formulas and arithmetic fluctuations

Arithmetic structure in a complete atlas of power-residue runs below $2^{32}$ : Rabung profiles, Kummer densities, norm dependencies, and local codes