Ljunggren--Jacobsthal and Bailey-Type Congruences for Rectangular Gaussian Binomial Coefficients

Technical Summary: Ljunggren–Jacobsthal and Bailey-Type Congruences for Rectangular Gaussian Binomial Coefficients

Problem and Motivation
The paper investigates arithmetic properties of a two-dimensional factorial ratio over the Gaussian integers $\mathbb{Z}[i]$, termed the rectangular Gaussian binomial coefficient, denoted $R(A, B; C, D)$. This object is defined as the quotient of a product over a translated $C \times D$ lattice rectangle by the product over the basic $C \times D$ rectangle:
$$R(A, B; C, D) := \frac{\prod_{x=A-C+1}^{A} \prod_{y=B-D+1}^{B} (x + iy)}{\prod_{x=1}^{C} \prod_{y=1}^{D} (x + iy)}.$$
The work is motivated by Kalinin's Gaussian analogue of Wolstenholme's theorem, which relates symmetric functions of reciprocal roots in $\mathbb{Z}[i]$ to high-power congruences. The author seeks to extend classical binomial congruences—specifically the Ljunggren–Jacobsthal congruence (concerning dilation of parameters by prime powers) and Bailey's refinement of Lucas's theorem (concerning decomposition into large blocks and digits)—to this two-dimensional Gaussian setting.

Methodology
The proofs rely on a combination of $p$-adic analysis in the ring of integers $\mathcal{O}_p = \mathbb{Z}_p[i]$ and combinatorial block decomposition. The analysis is restricted to rational primes $p \equiv 3 \pmod 4$, which are inert in $\mathbb{Z}[i]$. This condition ensures that $p$ remains prime in $\mathbb{Z}[i]$, the residue field is $\mathbb{F}_{p^2}$, and divisibility by $p$ in $\mathbb{Z}[i]$ corresponds to simultaneous divisibility of real and imaginary parts.

Key methodological steps include:

1. Reciprocal Power Sums: The author defines sums $H_j(k) = \sum_{z \in U_k} z^{-j}$ over complete sets of units $U_k$ modulo $p^k$. Using Kalinin's results and inductive arguments involving Newton's identities, the paper establishes that $H_1(k) \equiv 0 \pmod{p^{2k}}$ and $H_2(k) \equiv 0 \pmod{p^k}$ for $p > 5$.
2. Block Decomposition: Large rectangular products are partitioned into complete $p^k \times p^k$ blocks of units and boundary strips. The invariance of the product over a translated block is shown to depend on the vanishing of the first two reciprocal power sums.
3. Inclusion-Exclusion: The rectangular coefficient is expressed via multiplicative inclusion-exclusion of rectangular factorials $P(X, Y)$. This allows the separation of factors divisible by $p$ (which reduce to lower-scale coefficients) from the unit factors (which form the complete blocks).

Key Contributions and Results

1. p-Integrality: The paper first proves that $R(A, B; C, D)$ is $p$-integral (lies in $\mathcal{O}_p$) for all $p \equiv 3 \pmod 4$, establishing the well-definedness of the congruences.

2. Ljunggren–Jacobsthal-Type Supercongruence (Theorem 1.2):
   For a prime $p > 5$ with $p \equiv 3 \pmod 4$ and any integer $k \ge 1$, the simultaneous dilation of all parameters by $p$ yields:
   $$R(p^k A, p^k B; p^k C, p^k D) \equiv R(p^{k-1} A, p^{k-1} B; p^{k-1} C, p^{k-1} D) \pmod{p^{3k}}.$$
   This result proves the inert-prime case of a conjecture by Kalinin and generalizes the classical Ljunggren–Jacobsthal congruence to the Gaussian rectangular setting.

3. Bailey-Type Congruence (Theorem 1.3):
   For parameters decomposed into a large $p$-multiple and a residual digit ($A = pA' + \alpha$, etc.), the paper establishes a congruence modulo $p$:
   $$R(pA + \alpha, pB + \beta; pC + \gamma, pD + \delta) \equiv R(A, B; C, D) R(\alpha, \beta; \gamma, \delta) \binom{\alpha}{\gamma}^D \binom{\beta}{\delta}^C \pmod p.$$
   Unlike the one-dimensional Bailey congruence, this result includes two additional ordinary binomial factors, $\binom{\alpha}{\gamma}^D$ and $\binom{\beta}{\delta}^C$, which arise from the vertical and horizontal boundary strips of the rectangular decomposition.

4. Conceptual Framework:
   The paper situates these results within a "classical–Gaussian dictionary," mapping the integer lattice $\mathbb{Z}$ and circular functions to the square lattice $\mathbb{Z}[i]$ and lemniscatic functions (associated with the CM elliptic curve $y^2 = x^3 - x$). The symmetry group of the lattice (quarter-turn symmetry for $\mathbb{Z}[i]$ vs. reflection for $\mathbb{Z}$) dictates which reciprocal moments vanish, explaining the specific powers of $p$ in the congruences.

Significance and Scope
The paper claims to resolve the inert-prime case of Kalinin's conjecture regarding the Gaussian analogue of Wolstenholme-type congruences. It demonstrates that the arithmetic of rectangular Gaussian binomial coefficients is governed by the same mechanisms as classical binomial coefficients (reciprocal power sums and block invariance), albeit with structural modifications due to the two-dimensional lattice and the specific symmetry of $\mathbb{Z}[i]$.

The author explicitly notes limitations and future directions:

• The results are currently restricted to inert primes ($p \equiv 3 \pmod 4$). The split case ($p \equiv 1 \pmod 4$) requires a different treatment due to zero divisors in the residue ring.
• The Bailey-type congruence is proven only modulo $p$; higher-power lifts are suggested as a natural continuation but not proved.
• The paper proposes extensions to general imaginary quadratic orders and questions regarding global integrality and Barnes double-Gamma representations.

The work does not claim experimental applications but rather establishes a theoretical foundation for the arithmetic of Gaussian factorial ratios, linking them to the theory of CM elliptic curves and formal groups.