Dwork-type supercongruences through a creative q-microscope

We develop an analytical method to prove congruences of the type Sigma((pr-1)/d)(k=0) A(k)z(k) omega(z) Sigma((pr-1-1)/d)(k=0) A(k)z(pk) (mod p(mr)Z(p)[[z]]) for r = 1, 2, ..., for primes p > 2 and fixed integers m, d >= 1, where f(z) = E-k=0(infinity) A(k)z(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 >= 2 (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. (C) 2020 The Author(s). Published by Elsevier Inc.