A modular supercongruence for ${}_6{F}_5$: an Apéry-like story

Abstract

We prove a supercongruence modulo $p^3$ between the $p$th Fourier coefficient of a weight 6 modular form and a truncated ${}_6{F}_5$-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to $\zeta (3)$ to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence between the Apéry numbers and another Apéry-like sequence.

17 pages, to appear in Annales de l'Institut Fourier (Grenoble)

References in corpus (2)

• Extending Gaussian hypergeometric series to the $p$-adic setting
• Supercongruences and Complex Multiplication

Cited by in corpus (8)

• Supercongruences for rigid hypergeometric Calabi--Yau threefolds
• $p$-adic analogues of hypergeometric identities
• A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds
• Interpolated sequences and critical $L$-values of modular forms
• The Explicit Hypergeometric-Modularity Method I
• Supercongruences concerning truncated hypergeometric series
• On the divisibility of some truncated hypergeometric series
• Some Numeric Hypergeometric Supercongruences