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

Abstract

We develop an analytical method to prove congruences of the type $$\sum _{k=0}^{(p^r-1)/d}{A}_k{z}^k\equiv \omega (z)\sum _{k=0}^{(p^{r-1}-1)/d}{A}_k{z}^{pk}(\mathrm{mod}{p}^{mr}{\mathbb{Z}}_p[[z]])\text{for}r=1,2,\dots ,$$ for primes $p>2$ and fixed integers $m,d\ge 1$, 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 $m\ge 2$ (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 $r$.

34 pages

References in corpus (7)

• Congruences on sums of $q$-binomial coefficients
• Congruences for $q$-binomial coefficients
• Congruences concerning Legendre polynomials
• On a $q$-deformation of modular forms
• A common $q$-analogue of two supercongruences
• The method of creative microscoping
• A $q$-Dwork-type generalization of Rodriguez-Villegas' supercongruences

Cited by in corpus (5)

• $q$-Supercongruences from Jackson's ${}_8{\varphi }_7$ summation and Watson's ${}_8{\varphi }_7$ transformation
• ($q$-)Supercongruences hit again
• Some $q$-congruences involving central $q$-binomial coefficients
• $N$-tuple sum analogues for Ramanujan-type congruences
• q-analogues of the G.2 Supercongruence of Van Hamme