Split-prime supercongruence at the mixed CM point (1/6, 1/3; 1)

TL;DR

This paper proves a supercongruence for a hypergeometric coefficient sequence at split primes, revealing a CM-related enhancement and establishing an inert-prime obstruction, using modular and Cartier identity techniques.

Contribution

It establishes a new supercongruence for mixed CM hypergeometric coefficients at split primes, exceeding the generic weight prediction, and analyzes inert primes with formal and coefficient-level obstructions.

Findings

Proves A_{mp}^{mix} == A_m^{mix} mod p^4 for split primes p >= 7, p == 1 mod 3.

Identifies a CM enhancement factor of p in the supercongruence.

Establishes an inert-prime obstruction as a formal-parameter congruence and Cartier parity law.

Abstract

For the mixed CM point (a,b,c) = (1/6, 1/3, 1), define A_n^{mix} := 108^n [z^n] _2F_1(1/6, 1/3; 1; z)^3. For every split prime p >= 7, p == 1 mod 3, and every m >= 1, we prove unconditionally A_{mp}^{mix} == A_m^{mix} mod p^4. The exponent 4 exceeds the generic weight-3 Hodge-gap prediction of 3; the extra factor of p is a CM enhancement attached to j=0. We also establish the matching unconditional inert-prime obstruction (p == 2 mod 3), both as a formal-parameter congruence on the q-side and as a coefficient-level Cartier parity law modulo p. The proof uses the modular realization on Gamma_0(3) with parameter t = u/(1+27u)^2, a Lagrange-Burmann reduction to three Cartier identities Lambda_p(C_mix U_p^l) == 0 mod p^4 for l = 1,2,3, a saturated weak q-expansion lattice on the rigidified stack X_0(3) handling vertical integrality, and a length-three Witt-Cartier pole estimate at the elliptic point P_- driven by mu_3-equivariance of the canonical Frobenius lift.