\((q)\)-supercongruences hit again. (English) Zbl 1472.11029

Zudilin, Wadim

Hardy-Ramanujan J. 43, 46-55 (2020).

Summary: Using an intrinsic \(q\)-hypergeometric strategy, we generalise Dwork-type congruences \(H(p^{s+1})/H(p^s)\equiv H(p^s)/H(p^{s-1}) \pmod{p^3}\) for \(s=1,2,\ldots\) and \(p\) a prime, when \(H(N)\) are truncated hypergeometric sums corresponding to the periods of rigid Calabi-Yau threefolds.

MSC:

11A07	Congruences; primitive roots; residue systems
11B65	Binomial coefficients; factorials; \(q\)-identities
11F33	Congruences for modular and \(p\)-adic modular forms
33C20	Generalized hypergeometric series, \({}_pF_q\)
33D15	Basic hypergeometric functions in one variable, \({}_r\phi_s\)

Keywords:

hypergeometric sum; Ramanujan’s mathematics; supercongruence; \(q\)-congruence; creative microscoping; modular form; rigid Calabi-Yau threefold