A local-global theorem for p-adic supercongruences
Abstract Let ℤ p {\mathbb{Z}_{p}} denote the ring of all p-adic integers and call 𝒰 = { ( x 1 , … , x n ) : a 1 ⁢ x 1 + … + a n ⁢ x n + b = 0 } {\mathcal{U}}=\{(x_{1},\dots,x_{n}):a_{1}x_{1}+\dots+a_{n}x_{n}+b=0\} a hyperplane over ℤ p n {\mathbb{Z}_{p}^{n}} , where at least one of a 1 , … , a n {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 {p^{r}} over some suitable collection of r hyperplanes, then it is zero modulo p r {p^{r}} over the whole ℤ p n {\mathbb{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: ∑ k = 0 p - 1 ( 2 5 ) k 5 ( k ! ) 5 ≡ - Γ p ⁢ ( 1 5 ) 5 ⁢ Γ p ⁢ ( 2 5 ) 5 ⁢ ( mod ⁢ p 5 ) \sum_{k=0}^{p-1}\frac{(\frac{2}{5})_{k}^{5}}{(k!)^{5}}\equiv-\Gamma_{p}\bigg{(% }\frac{1}{5}\bigg{)}^{5}\Gamma_{p}\bigg{(}\frac{2}{5}\bigg{)}^{5}~{}(\mathrm{% mod}\,p^{5}) for each p ≡ 1 ⁢ ( mod ⁢  5 ) {p\equiv 1~{}(\mathrm{mod}\,5)} , where ( x ) k = x ⁢ ( x + 1 ) ⁢ ⋯ ⁢ ( x + k - 1 ) {(x)_{k}=x(x+1)\cdots(x+k-1)} and Γ p {\Gamma_{p}} denotes the p-adic Gamma function.