Supercongruences for truncated hypergeometric series with applications to certain weight three newforms

We prove general results on supercongruences between values of truncated hypergeometric functions and their character analogs. As a consequence of the main results of this paper, we prove Beukers-type supercongruences for certain weight three newforms.

     

Supercongruences between truncated hypergeometric functions and their Gaussian analogs

Fernando Rodriguez-Villegas has conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension . For manifolds of dimension , he observed four potential supercongruences. Later the author proved one of the four. Motivated by Rodriguez-Villegas's work, in the present paper we prove a general result on supercongruences between values of truncated hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, we prove the three remaining supercongruences.

     

On a Supercongruence Conjecture of Z.-W. Sun

In this paper, the author partly proves a supercongruence conjectured by Z.-W.Sun in 2013. Let p be an odd prime and let a ∈ Z⁺. Then, if p ≡ 1 (mod 3),k=0 6p^a 2k k/16^k ≡ 3/p^a(modp²) is obtained, where is the Jacobi symbol.