A $q$-analogue of a binomial supercongruence

Abstract

In a recent paper arXiv:2503.20727, Ichino and Prasanna proved a supercongruence for binomial coefficients and related it to representations of ${\mathrm{GL}}_2$ over $p$-adic rings. We prove a $q$-analog of this supercongruence using the $q$-Lucas theorem, from which a strengthening of the main result of loc. cit. can easily be deduced. Our proof also substantially simplifies the original argument of arXiv:2503.20727.

10 pages