A local-global theorem for p-adic supercongruences

Let Z(p) denote the ring of all p-adic integers and callu = {(x(1), ..., x(n)) : a(1)x(1) + ... + a(n)x(n) + b = 0}a hyperplane over Z(p)(n), where at least one of a(1), ...,an 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 Z(p)(n). We provide various applications of this general criterion by establishing several p-adic analogues of hypergeometric identities. For example, we confirm a conjecture of Deines et al. as follows:Sigma(k=0)(5)(p-1)2/4)k)/(k!)(5) = -Gamma(p) (1/5)(5) Gamma(p) (2/5)(5) (mod p(5))for each p = 1 (mod 5), where (x)(k) = x(x + 1) ... (x + k - 1) and Gamma(p) denotes the p-adic Gamma function.