Dwork congruences via q-deformation

TL;DR

This paper proves a q-deformation of Dwork's congruences for polynomials arising from K-theoretic vertex functions of cotangent bundles over Grassmannians, connecting to p-adic congruences.

Contribution

It introduces a q-deformation of Dwork's congruences for specific polynomials related to Grassmannian geometry, extending previous p-adic results.

Findings

Polynomials satisfy a q-deformed Dwork congruence relation.

In the limit q→1, recovers classical Dwork congruences.

Connects K-theoretic vertex functions with p-adic congruences.

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