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.