Dwork-type supercongruences through a creative $q$-microscope

Abstract

We develop an analytical method to prove congruences of the type $$ \sum_{k=0}^{(p^r-1)/d}A_kz^k \equiv \omega(z)\sum_{k=0}^{(p^{r-1}-1)/d}A_kz^{pk} \pmod{p^{mr}\mathbb Z_p[[z]]} \quad \text{for}\; r=1,2,\dots, $$ for primes $p>2$ and fixed integers $m,d\ge1$, where $f(z)=\sum_{k=0}^\infty A_kz^k$ is an "arithmetic" hypergeometric series. Such congruences for $m=d=1$ were introduced by Dwork in 1969 as a tool for $p$-adic analytical continuation of $f(z)$. Our proofs of several Dwork-type congruences corresponding to $m\ge2$ (in other words, supercongruences) are based on constructing and proving their suitable $q$-analogues, which in turn have their own right for existence and potential for a $q$-deformation of modular forms and of cohomology groups of algebraic varieties. Our method follows the principles of creative microscoping introduced by us to tackle $r=1$ instances of such congruences; it is the first method capable of establishing the supercongruences of this type for general $r$.