Order drop, Hecke descent, and a mod $p^4$ supercongruence for symmetric-cube hypergeometric coefficients

Abstract

We prove that the symmetric-cube coefficients $A_n=(-27)^n[z^n]\,_2F_1(1/3,1/3;1;z)^3$ satisfy the supercongruence $A(mp)\equiv A(m) \bmod p^4$ for every prime $p\geq 5$ and every $m\geq 1$. The proof rests on three ingredients: (i) the modular identification $F(t(\tau))=\eta(\tau)^9/\eta(3\tau)^3$ with $t(\tau)=\eta(3\tau)^{12}/\eta(\tau)^{12}$, whose logarithmic derivative is the weight-5 Eisenstein series $C(q)=3E_5(\chi_0,\chi_3)$ on $\Gamma_0(3)$; (ii) exact congruences $c_{mp^r}\equiv c_{mp^{r-1}} \bmod p^{4r}$ for the coefficients of $C$, combined with a Lagrange-Burmann extraction; and (iii) a Hecke descent on weakly holomorphic forms, where the defect is expanded in the two-dimensional space of weight-5 forms on $\Gamma_0(3)$ with character $\chi_3$, spanned by $C$ and $tC$, via a cusp-adapted basis, with the second cusp handled by the Fricke involution $W_3$. As an independent result, we show that the Mao-Tian cubic recurrence drops from order 3 to order 2 at the specialization $(1/3,1/3,1)$.