曲线$X^a=Y^b$的点计数的模性：新的Rogers–Ramanujan恒等式
Modularity of Point Counts for the Curves $X^a=Y^b$: New Rogers--Ramanujan Identities
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中文总结 AI 辅助
该研究针对互素$1<a<b$的曲线$X^a=Y^b$的点计数猜想，证明$a=3$层的新Rogers–Ramanujan恒等式，且由AxiomProver在Lean中验证。
中文摘要 AI 辅助
对于互素的$1<a<b$，设$M_n^{a,b}(\boldsymbol{F}_q)$为有限域$\boldsymbol{F}_q$上满足$X^a=Y^b$的$n\times n$幂零矩阵的可交换对集合。Huang、Jiang和Oblomkov将其阶数组装成欧拉$q$-级数$Z_{a,b}(q)$，他们猜想该级数是包含雅可比θ函数和戴德金η函数的显式乘积$P_{a,b}(q)$，这蕴含三重等式：点计数项$\boldsymbol{\text{point count}}$即$\boldsymbol{\text{point count}}$等于$q$-级数$Z_{a,b}(q)$，也等于θ商$P_{a,b}(q)$。若猜想成立，则曲线$X^a=Y^b$上的点计数本质上是$\boldsymbol{\text{point count}}$在$\boldsymbol{\text{point count}}$上的模函数。该猜想按$a$分层，每个$b$对应一个恒等式，$a=2$层是经典结果，包含Rogers–Ramanujan和Andrews–Gordon恒等式，而$a\boldsymbol{\text{point count}}$的$a=3$层：一个新的无穷族Rogers–Ramanujan恒等式，以及Warnaar乘积的几何起源。AxiomProver在Lean中基于现有文献验证了这些新恒等式。
英文摘要
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 $Γ(a+b)$. 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.
