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

Abstract

We prove that the symmetric-cube coefficients $A_n=(-27{)}^n[z^n]{}_2{F}_1(1/3,1/3;1;z{)}^3$ satisfy the supercongruence $A(mp)\equiv A(m)\mathrm{mod}{p}^4$ for every prime $p\ge 5$ and every $m\ge 1$. 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 $c_{m{p}^r}\equiv c_{m{p}^{r-1}}\mathrm{mod}{p}^{4r}$ 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 $(1/3,1/3,1)$.

25 pages. v2: substantially revised proof. The earlier three-layer truncation argument is replaced by a cusp-adapted expansion in the weight-5 modular forms space on ${\Gamma }_0(3)$ with character ${\chi }_3$, giving $F_r\equiv 0\mathrm{mod}{p}^4$ for all $r\ge 1$ uniformly. The order drop from 3 to 2 is now stated as an independent result and is not used in the proof of Theorem A