Deformation and Symmetry in Two Hypergeometric Supercongruences

Abstract

We prove two supercongruence conjectures of Li--Long--Tu concerning truncated very-well-poised ${}_7F_6(1)$ series and Fourier coefficients of modular forms. The proofs are based on a common deformation method. By deforming the original truncated hypergeometric series to suitable terminating series, we are able to apply classical hypergeometric transformations together with Mishev's $W(D_5)$ symmetry. This reduces both problems to congruences for lower-rank hypergeometric sums that can be treated using the $p$-adic methods of Long--Tu--Yui--Zudilin. Although the two cases require different local arithmetic calculations, their global proof structure is the same. The resulting argument gives a unified explanation of these two supercongruences and suggests that the same deformation-and-symmetry method may apply more broadly.