A full $p^{4r}$ supercongruence tower for a level-three symmetric-cube hypergeometric sequence

Technical Summary: A Full $p^{4r}$ Supercongruence Tower for a Level-Three Symmetric-Cube Hypergeometric Sequence

Problem Statement
The paper addresses the arithmetic properties of the coefficients $A_n$ in the expansion of the cube of a specific hypergeometric series:
$$\left[ {}_2F_1\left(\frac{1}{3}, \frac{1}{3}; 1; 27z\right) \right]^3 = \sum_{n \ge 0} A_n z^n.$$
While the sequence $A_n$ and its modular parametrization on the modular curve $X_0(3)$ were previously known (specifically via the work of Moy), the paper seeks to establish a "full prime-power supercongruence tower." Specifically, it aims to prove that for any prime $p \ge 5$ and integers $m, r \ge 1$, the coefficients satisfy the congruence:
$$A_{mp^r} \equiv A_{mp^{r-1}} \pmod{p^{4r}}.$$
Previous results had established this congruence only for the case $r=1$ (modulo $p^4$) or for ordinary congruences modulo $p^r$. The challenge lies in extending the modulus from $p^4$ to $p^{4r}$ for all prime powers $r \ge 1$.

Methodology
The proof is entirely modular, relying on the theory of modular forms, Hecke operators, and the geometry of the curve $X_0(3)$. The methodology proceeds through the following steps:

1. Modular Parametrization: The author defines $B_n = (-1)^n A_n$ and realizes the generating function $\sum B_n t^n$ as a modular form $\Theta(\tau) = \eta(\tau)^9 / \eta(3\tau)^3$ on $X_0(3)$, where $t(\tau) = \eta(3\tau)^{12} / \eta(\tau)^{12}$ is a Hauptmodul.
2. Logarithmic Derivative and Eisenstein Series: The logarithmic derivative of the generating function, denoted $C(q)$, is identified as the Eisenstein series $3E_{5, \chi_0, \chi_3}$ of weight 5 and character $\chi_3$.
3. Coefficient Extraction: Using the Lagrange–Buermann formula, the coefficients $B_m$ are expressed as the constant term (coefficient of $q^0$) of the Laurent series $C(q)/t(q)^m$.
4. Defect Analysis: To prove the congruence, the author analyzes the "defect" between terms at different prime powers. Instead of iterating a one-prime Cartier operator, the paper introduces a sparse Cartier defect $F_{m,s}$ and a corresponding modular Hecke lift $G_{m,s}$.
   • The defect is defined as $F_{m,s} = \Lambda_p^{s+1} f_{mp^{s+1}} - \Lambda_p^s f_{mp^s}$, where $f_N = C/t^N$ and $\Lambda_p$ is the Cartier operator.
   • The modular lift is $G_{m,s} = T_{p^{s+1}} f_{mp^{s+1}} - T_{p^s} f_{mp^s}$, where $T_{p^k}$ are prime-power Hecke operators.
5. Inductive Argument on Hecke Layers: The proof proceeds by induction on the prime-power exponent $s$. The core idea is to show that $G_{m,s}$ is divisible by $p^{4(s+1)}$.
   • Hecke Expansion: Proposition 5.1 provides an explicit expansion of $T_{p^a}$ in terms of $\Lambda_p$ and $V_p$ (the Frobenius operator), showing that $G_{m,s}$ differs from the sparse defect $F_{m,s}$ by terms divisible by high powers of $p$, assuming the inductive hypothesis.
   • Modular Space Analysis: The form $G_{m,s}$ is shown to belong to the space of weakly holomorphic modular forms $M_5^!(\Gamma_0(3), \chi_3)$. This space is spanned by $\{C, tC\}$.
   • Fricke Involution: A crucial step involves applying the Fricke involution $W_3$. The author demonstrates that $G_{m,s}|W_3$ has a high order of vanishing at the cusp $\infty$ (specifically $\ge m+1$). This forces the "principal part" coefficients of $G_{m,s}$ (the coefficients of $C/t^j$ for $j < m$ and the $tC$ term) to vanish exactly, while the remaining coefficients are shown to be divisible by $p^{4(s+1)}$ via the inductive hypothesis.

Key Contributions

• Extension of Modulus Depth: The primary contribution is the extension of the supercongruence modulus from $p^4$ (the depth-one case proved in a previous work by the author) to $p^{4r}$ for all $r \ge 1$.
• Prime-Power Hecke Defects: The paper replaces the "one-prime Hecke defect" used in previous literature with a "prime-power defect." This allows for the iteration of the congruence proof to arbitrary prime powers.
• Sparse Cartier Defects: The introduction of the sparse Cartier defect $F_{m,s}$ serves as the bridge between the modular forms and the arithmetic coefficients, isolating the specific terms that control the congruence.
• Exact Vanishing via Fricke Involution: The proof utilizes the Fricke involution on the two-dimensional space $M_5(\Gamma_0(3), \chi_3)$ to kill the low-order terms exactly, a technique that strengthens the result beyond what formal differential congruences alone could achieve.

Results
The paper proves Theorem 1.1: For every prime $p \ge 5$ and all integers $m, r \ge 1$,
$$A_{mp^r} \equiv A_{mp^{r-1}} \pmod{p^{4r}}.$$
This result confirms that the sequence $A_n$ satisfies a "full $p^{4r}$ supercongruence tower." The proof establishes that the difference between the coefficients at indices $mp^r$ and $mp^{r-1}$ is divisible by $p^{4r}$.

Significance and Claims
The paper explicitly states that the sequence $A_n$, the modular parametrization involving $\eta$-quotients, and the ordinary $p^r$-level congruences were already recorded by Moy. The novelty of this work is strictly the depth-preserving strengthening of the modulus.

• The author claims that the ordinary Moy framework yields congruences modulo $p^r$, but the additional precision of $p^{4r}$ is not produced by formal differential congruences alone.
• The result is achieved by recovering the extra precision $p^{3r}$ through the combination of prime-power Hecke expansions and the Fricke order drop.
• The paper does not claim new applications or experimental proposals; its significance is purely theoretical, contributing to the "level-three dictionary" of supercongruences by providing a complete tower for this specific symmetric-cube hypergeometric sequence.