Multiple Rogers-Ramanujan type identities for inert quadratic orders
Shane Chern, Yifeng Huang
TL;DR
The paper analyzes finitized Coh zeta and Quot zeta functions for inert quadratic orders $R'_{2,2m}$ over ${\mathbb F}_q$, deriving a $2m$-fold multisum that converges to an $m$-fold Bressoud sum and proving a conjecture by the second author. A key innovation is the introduction of a ghost parameter $a$ in the summands, followed by a purely $q$-theoretic demonstration that the resulting sum is independent of $a$, which enables a master interpolation unifying the three quadratic orders. The authors connect detailed lattice-module combinatorics (boundary lattices, totally real and co-totally-real submodules) to explicit $q$-series expressions, ultimately proving that the inert case matches the corresponding Bressoud-type identity and refining the framework via a master polynomial. The work reveals a deep link between higher-rank Donaldson–Thomas-type enumerations for plane curve singularities and classical multiple Rogers–Ramanujan identities, and proposes a reflective and cyclic-sieving structure that hints at a geometric/combinatorial underpinning. Overall, the results provide a comprehensive arithmetic-analytic picture of Quot and Coh zeta functions across ramified, split, and inert quadratic orders, with a unified $q$-theoretic and combinatorial interpretation of the arising identities.
Abstract
We compute the Quot and finitized Coh zeta functions of the inert quadratic orders $\mathbb{F}_q[[T]]+T^{m}\mathbb{F}_{q^{2}}[[T]]$ for every $m\geq 1$ in terms of a $2m$-fold multisum, and then show this multisum equals an $m$-fold Bressoud sum. This proves a recent conjecture of the second author, rounding up the line of exploration in the series of work by the authors and Jiang. The equality between the $2m$-fold multisum and the $m$-fold Bressoud sum is built upon generalizing the multisum by introducing a ``ghost'' parameter $a$ to its summands. We then show that such an $a$-generalization is surprisingly $a$-independent by purely $q$-theoretic techniques. Finally, we propose a refined multisum that interpolates two versions of Quot zeta functions for all three types of quadratic orders.