Dwork-type supercongruences through a creative q-microscope

Abstract

We develop an analytical method to prove congruences of the type∑k=0(pr−1)/dAkzk≡ω(z)∑k=0(pr−1−1)/dAkzpk(modpmrZp[[z]])forr=1,2,…, for primes p>2 and fixed integers m,d⩾1, where f(z)=∑k=0∞Akzk 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.

Topics & Concepts