Dwork congruences via q-deformation

TL;DR

The paper constructs polynomials $T_s(z,q)\in\mathbb{Z}[z,q]$ as truncations of the K-theoretic vertex function for the Nakajima variety $X=T^*Gr(k,n)$ and proves a $q$-deformed Dwork congruence: $\dfrac{T_{s+1}(z,q)}{T_s(z^p,q^p)} \equiv \dfrac{T_s(z,q)}{T_{s-1}(z^p,q^p)} \pmod{[p^s]_q}$. In the limit $q\to 1$, these reduce to the congruences established by Narain–Varchenko/Schneider–Vologodsky for the truncations in SV, thereby unifying arithmetic and $q$-deformed geometric contexts. The proof employs a root-of-unity reduction and a multiplicative identity mirroring Dwork’s mechanism, framing a $q$-deformed unit-root analogue $\lambda(z,q)$. The results illuminate how $q$-deformations of K-theoretic vertex functions encode congruence phenomena akin to zeta-function arithmetic, with potential connections to $q$-Frobenius intertwiners in the $q$-hypergeometric setting.

Abstract

We consider a system of polynomials $T_{s}(z,q)\in\mathbb{Z}[z,q]$ which appear as truncations of the K-theoretic vertex function for the cotangent bundles over Grassmannians $T^{*}Gr(k,n)$. We prove that these polynomials satisfy a natural $q-$deformation of Dwork's congruences \[\frac{T_{s+1}(z,q)}{T_{s}(z^{p},q^{p})}\equiv\frac{T_{s}(z,q)}{T_{s-1}(z^{p},q^{p})}\text{ (mod } [p^{s}]_{q})\] In the limit $q\to 1$ we recover the main result of arXiv:2302.03092v3

Figures (1)

• Figure 1: $A_{n-1}$ quiver with two framings

Theorems & Definitions (6)

• Theorem 1.1
• Theorem 2.1
• Theorem 3.1
• proof
• Corollary 3.2
• proof