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

Victor J. W. Guo; Wadim Zudilin

doi:10.1016/j.jcta.2020.105362

paper

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

arXiv:2001.02311 · doi:10.1016/j.jcta.2020.105362

Abstract

We develop an analytical method to prove congruences of the type $k = 0 \sum (p^{r} - 1) / d A_{k} z^{k} \equiv ω (z) k = 0 \sum (p^{r - 1} - 1) / d A_{k} z^{p k} (mod p^{m r} Z_{p} [[z]]) for r = 1, 2, \dots,$ for primes $p > 2$ and fixed integers $m, d \geq 1$ , where $f (z) = \sum_{k = 0}^{\infty} A_{k} z^{k}$ 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 $m \geq 2$ (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 $r$ .

arXiv comment34 pages

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

Abstract

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Dwork-type supercongruences through a creative q-microscope

Abstract

References in corpus (16)

Cited by in corpus (6)

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