A local-global theorem for $p$-adic supercongruences

Abstract

Let ${\mathbb{Z}}_p$ denote the ring of all $p$-adic integers and call $$\mathcal{U}=\{(x_1,\dots ,x_n):a_1{x}_1+\dots +a_n{x}_n+b=0\}$$ a hyperplane over ${\mathbb{Z}}_p^n$, where at least one of $a_1,\dots ,a_n$ is not divisible by $p$. We prove that if a sufficiently regular $n$-variable function is zero modulo $p^r$ over some suitable collection of $r$ hyperplanes, then it is zero modulo $p^r$ over the whole ${\mathbb{Z}}_p^n$. We provide various applications of this general criterion by establishing several $p$-adic analogues of hypergeometric identities.

45 pages. This is a preliminary manuscript. Some new congruences are added