Modularity of Point Counts for the Curves $X^a=Y^b$: New Rogers--Ramanujan Identities

Technical Summary: Modularity of Point Counts for the Curves $X^a = Y^b$: New Rogers–Ramanujan Identities

Problem Statement
The paper addresses a conjecture by Huang, Jiang, and Oblomkov (HJO) concerning the enumerative geometry of the singular plane curve $X^a = Y^b$ over finite fields. For coprime integers $1 < a < b$, let $M_{n}^{a,b}(\mathbb{F}_q)$ denote the set of commuting pairs of nilpotent $n \times n$ matrices over $\mathbb{F}_q$ satisfying $X^a = Y^b$. HJO assembled the normalized point counts of these sets into a formal power series $Z_{a,b}(q)$. They conjectured that $Z_{a,b}(q)$ equals an explicit infinite product $P_{a,b}(q)$ involving Jacobi theta functions and Dedekind's eta-function.

If true, this equality implies that the point counts on the curve $X^a = Y^b$ are governed by modular functions on the congruence subgroup $\Gamma(a+b)$. The conjecture generalizes the classical Rogers–Ramanujan (RR) and Andrews–Gordon (AG) identities. Specifically:

• For $a=1$, the identity is trivial.
• For $a=2$, the conjecture recovers the known AG identities (including the classical RR identities for $b=3$).
• For $a \ge 3$, the conjecture remained unproven.

The primary objective of this paper is to prove the conjecture for the $a=3$ layer, establishing a new infinite family of Rogers–Ramanujan type identities and providing a geometric origin for products previously studied by Warnaar.

Methodology
The authors employ a multi-step strategy bridging algebraic geometry, combinatorics, and representation theory:

1. Geometric Setup and Gap Sets: The problem is translated into the language of numerical semigroups. The gap set $G_{a,b} = \mathbb{N} \setminus \langle a, b \rangle$ is identified with a Young diagram within an $a \times b$ rectangle. For $a=3$, this gap set consists of two interlacing arithmetic progressions. The series $Z_{3,b}(q)$ is defined as a sum over integer vectors indexed by this gap set, weighted by a quadratic form $Q_{3,b}(n)$ and $q$-Pochhammer symbols.

2. Reindexing and Coordinate Adaptation: The authors introduce coordinates adapted to the two arithmetic progressions comprising $G_{3,b}$. By fixing a "boundary chain" of variables (corresponding to the outer boundary of the Young diagram), the sum is decomposed. The quadratic form $Q_{3,b}(n)$ is re-expressed in terms of these new coordinates, separating the contribution into a boundary term and a finite Laurent polynomial involving Gaussian polynomials (q-binomial coefficients).

3. Connection to $q$-Supernomials: The core technical innovation is recognizing the finite Laurent polynomial arising from the fixed-boundary sum as a specific coefficient of a completely symmetric $q$-supernomial. The authors utilize the Schilling–Warnaar theory of $q$-supernomials, specifically:

   • Multitableau Expansion: The sum is interpreted as a generating function for tuples of one-row semistandard tableaux (entries $\{1, 2\}$).
   • Order Independence: A critical step involves showing that the sum is independent of the ordering of the two blocks of tableaux (corresponding to the two arithmetic progressions). This symmetry allows the authors to reconcile the two congruence classes $b \equiv 1 \pmod 3$ and $b \equiv -1 \pmod 3$ into a single identity.
   • Fermionic Expansion: The supernomial is transformed into a fermionic sum (a sum of products of Gaussian polynomials) using rigged configuration bijections.

4. Reassembly and Modular Identification: The fixed-boundary sums are reassembled to reconstruct the full series $Z_{3,b}(q)$. The resulting fermionic sums are identified as the sum sides of Warnaar's $A_2$ Andrews–Gordon identities. Finally, the authors perform a residue-class calculation to show that the resulting infinite products match the HJO conjectured product $P_{3,b}(q)$, which is expressed as a quotient of theta functions.

Key Contributions and Results
The paper proves the following main theorem (Theorem 1.1):

• Proof of the $a=3$ Conjecture: For every integer $b > 3$ coprime to 3, the normalized point count $P_{3,b}^{pt}(q)$, the series $Z_{3,b}(q)$, and the product $P_{3,b}(q)$ are equal.
• New Rogers–Ramanujan Identities: The equality establishes an infinite family of identities where a sum over a gap-set-indexed lattice (involving a quadratic form and Gaussian polynomials) equals a product of theta functions.
• Modularity: The series $Z_{3,b}(q)$ is shown to be a finite product of Jacobi theta quotients of level $b+3$. Consequently, it is a modular function with character on $\Gamma(b+3)$ (up to a rational power of $q$).
• Combinatorial Interpretation: The paper provides a tableau model (Corollary 1.4) for $Z_{3,b}(q)$. The sum side is interpreted as a weighted sum over tuples of one-row semistandard tableaux, where the weight involves a coinversion statistic and specific boundary conditions determined by $b$.
• Verification: The authors note that the identities were formally verified in the Lean proof assistant using the AxiomProver system, assuming existing literature on $q$-supernomials and theta functions.

Significance
The paper claims significance in several specific areas:

1. Geometric Origin of RR Phenomena: It provides a concrete geometric realization of Rogers–Ramanujan type identities via point counts on singular curves $X^a = Y^b$, moving beyond the traditional representation-theoretic origins (affine Lie algebras, Virasoro modules).
2. Resolution of the $a=3$ Layer: While the $a=2$ case was known (Andrews–Gordon), the $a \ge 3$ cases were previously unknown. This work fully resolves the $a=3$ case.
3. Unification of Structures: The proof demonstrates a bridge between the combinatorics of numerical semigroups (gap sets), the algebra of $q$-supernomials (Schilling–Warnaar theory), and the modular properties of theta functions.
4. Clarification of Warnaar's Products: The work provides a geometric origin for specific products found by Warnaar, linking them directly to the enumerative geometry of plane curves.

The authors maintain a modest tone regarding the scope, noting that while they prove the $a=3$ layer, the general case for $a \ge 4$ remains open. The work is presented as a rigorous verification of the HJO conjecture for a specific, non-trivial layer, utilizing established combinatorial machinery to solve a geometric enumeration problem.