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Dwork-type supercongruences through a creative \(q\)-microscope. (English) Zbl 1473.11046
After introducing the unit root \(\omega(z)\), a \(p\)-adic analytical function shaped as Dirichlet quadratic character, the authors illustrate the congruences \[ \sum\limits_{k=0}^{(p^r-1)/d} A_k z^k \equiv \omega(z) \sum_{k = 0}^{(p^{r-1}-1)/d} A_k z^{p k} \pmod {p^{mr} \mathbb{Z}_p [[z]]}\] as extension to \(r,m,d \in \mathbb{Z^+}\) of those found by B. Dwork [Publ. Math., Inst. Hautes Étud. Sci. 37, 27–115 (1969; Zbl 0284.14008)] for the specific case \(m=d=1\).
Then they clarify that \(f(z) = \sum_{k = 0}^\infty A_k z^k\) is an arithmetic hypergeometric series and they focus on the truncation, at \(z=1\), of the following two: \[ \sum_{k=0}^{\infty} (8k+1) \binom {4k}{2k} {\binom {2k}{k}}^2 \frac {z^k}{2^{8k}3^{2k}},\] \[ \sum_{k=0}^{\infty} \frac {\left( \frac{1}{2} \right)^3_k}{k!^3} (3k+1) (4z)^k ,\] corresponding to Dwork-type supercongruences (\(m>1\)) here established, respectively for primes p>3 and p>2, via the strategy of creative \(q\)-microscoping.
Supplied in a previous joint paper [Adv. Math. 346, 329–358 (2019; Zbl 1464.11028)], such method is aimed at proving supercongruences for truncated sums of arithmetic hypergeometric evaluations. Beyond \(q\)-congruences and \(q\)-identities successfully used on the same topic by the first author [J. Math. Anal. Appl. 487, Article 124022 (2020; Zbl 1439.11011)], the proof employs the transformation formulas of basic hypergeometric series available in the Encyclopedia of Mathematics and its Applications; namely, the Volume No. 96 [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)].
In addition, this paper establishes several \(q\)-analogues of Dwork-type congruences, some of which partly confirm the conjectural work of H. Swisher [Res. Math. Sci. 2, Paper No. 18, 21 p. (2015; Zbl 1337.33005)], and it provides new similar conjectures too.
Eventually, the authors suggest the investigation of certain Dwork-type \(q\)-congruences, connected to the modular Calabi-Yau threefold studied by S. Ahlgren and K. Ono [Monatsh. Math. 129, No. 3, 177–190 (2000; Zbl 0999.11031)], in order to explore the \(q\)-deformations proposed by P. Scholze [Ann. Fac. Sci. Toulouse, Math. (6) 26, No. 5, 1163–1192 (2017; Zbl 1461.14031)].

MSC:
11B65 Binomial coefficients; factorials; \(q\)-identities
05A30 \(q\)-calculus and related topics
11A07 Congruences; primitive roots; residue systems
33C05 Classical hypergeometric functions, \({}_2F_1\)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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