Representations of $\mathrm{GL}_2$ over $\mathbb{Z}/p^n\mathbb{Z}$ and supercongruences for hypergeometric polynomials

Abstract

For an odd prime $p$, we realize the trivial representation of $\mathrm{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ on the free $\mathbb{Z}/p^n \mathbb{Z}$-module of rank one as a subquotient of a direct sum of symmetric power representations (twisted by appropriate powers of the determinant) of rank strictly greater than one. The proof eventually reduces to establishing some novel supercongruences for hypergeometric polynomials.