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

Abstract

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 $$\underset{\text{point count}}{\underbrace{\prod _{n\ge 1}(1-q^n)\cdot \left(\sum _{n=0}^{\infty }\frac{|M_n^{a,b}({\mathbb{F}}_q)|}{|{\mathrm{GL}}_n({\mathbb{F}}_q)|}\right){|}_{q\mapsto q^{-1}}}}=\underset{q\text{-series}}{\underbrace{Z_{a,b}(q)}}=\underset{\text{theta quotient}}{\underbrace{P_{a,b}(q)}}$$ 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 $a\ge 3,$ 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.

Comments welcome