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

TL;DR

This paper establishes a local-global principle for $p$-adic functions, showing that if a regular function vanishes modulo $p^r$ on certain hyperplanes, it vanishes on the entire space, with applications to $p$-adic hypergeometric identities.

Contribution

It introduces a new criterion linking local hyperplane conditions to global vanishing in $p$-adic spaces, enabling derivation of $p$-adic hypergeometric identities.

Findings

Proves a local-global theorem for $p$-adic functions.

Derives several $p$-adic hypergeometric identities.

Provides a framework for $p$-adic analogues of classical identities.

Abstract

Let ${\mathbb Z}_p$ denote the ring of all $p$-adic integers and call $${\mathcal U}=\{(x_1,\ldots,x_n):\,a_1x_1+\ldots+a_nx_n+b=0\}$$ a hyperplane over ${\mathbb Z}_p^n$, where at least one of $a_1,\ldots,a_n$ 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.