Modularity of Point Counts for the Curves $X^a=Y^b$: New Rogers--Ramanujan Identities
For coprime $1<a<b$, let $M_n^{a,b}(\mathbb{F}_q)$ be the set of commuting pairs of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ with $X^a=Y^b$. Huang, Jiang, and Oblomkov assembled their orders as an Eulerian $q$-series $Z_{a,b}(q)$. They conjectured that it is an explicit product $P_{a,b}(q)$ involving Jacobi's theta function and Dedekind's eta-function, implying the threefold equality $$\underbrace{\prod_{n\ge1}(1-q^n)\cdot\Biggl(\sum_{n=0}^{\infty}\frac{|M_n^{a,b}(\mathbb{F}_q)|}{|\mathrm{GL}_n(\mathbb{F}_q)|}\Biggr)\Biggr|_{q\mapsto q^{-1}}}_{\text{point count}}\;=\;\underbrace{Z_{a,b}(q)}_{q\text{-series}}\;=\;\underbrace{P_{a,b}(q)}_{\text{theta quotient}}$$ If true, the point count on $X^a=Y^b$ is essentially a modular function on $\Gamma(a+b)$. The conjecture is layered in $a$, with an identity for each $b$. The $a=2$ layer is classical, including identities of Rogers--Ramanujan and Andrews--Gordon. For $a\geq3,$ nothing was known. We prove the $a=3$ layer in full: a new infinite family of Rogers--Ramanujan identities, and a geometric origin for Warnaar's products. AxiomProver verified these new identities in Lean assuming existing literature.