An extension of the Apéry number supercongruence. (English) Zbl 1170.11008

Let \[ f(z):= \eta^{4}(2z) \eta^{4}(4z)=\sum_{n=1}^{\infty} a(n)q^n, \] where \(q:=e^{2 \pi i z}\) and \[ \eta(z):=q^{\frac{1}{24}} \prod_{n=1}^{\infty}(1-q^n) \] is Dedekind’s eta function. Then \(f\) is the unique normalised eigenform in the space of weight four cusp forms on the congruence subgroup \(\Gamma_0(8)\). The main result of the paper is the following supercongruence. Theorem 1. Let \(p\) be an odd prime. Then \[ \sum_{j=0}^{p-1} {\binom{2j}{j}}^4 2^{-8j} \equiv a(p) \pmod {p^3}. \tag{1} \] This is one of 22 possible supercongruences identified by F. Rodriguez-Villegas [Hypergeometric families of Calabi-Yau manifolds, Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS). Fields Inst. Commun. 38, 223–231 (2003; Zbl 1062.11038)] while examining the relationship between the number of points over \(\mathbb{F}_p\) on certain Calabi-Yau manifolds and (truncated) hypergeometric series which correspond to a particular period of the manifold. Note that the left-hand side of (1) can be expressed as the truncated ordinary hypergeometric series \[ {{_{4}F_3} \left[ \begin{matrix} \frac{1}{2}, & \frac{1}{2}, & \frac{1}{2}, & \frac{1}{2}\;\\ & 1, & 1, & 1 \end{matrix} \Big| \; 1 \right]}_{p-1} =\sum^{p-1}_{j=0} \frac{\phi{\frac{1}{2}}{j}^4} {j!^4}. \] Also, for odd primes, \(a(p)\) is related to the number of points, \(N_p\), over \(\mathbb{F}_p\) on the modular Calabi-Yau threefold \[ x+\frac{1}{x}+y+\frac{1}{y}+z+\frac{1}{z}+w+\frac{1}{w}=0 \] via \[ a(p)=p^3-2p^2-7-N_p. \] [See S. Ahlgren and K. Ono, Monatsh. Math. 129, No. 3, 177–190 (2000; Zbl 0999.11031).]
The proof of Theorem 1 uses a result of S. Ahlgren and K. Ono [J. Reine Angew. Math. 518, 187–212 (2000; Zbl 0940.33002)], which relies on the above relationship, to express \(a(p)\) in terms of hypergeometric series over finite fields or Gaussian hypergeometric series. The author expands this series \(p\)-adically and uses the Gross-Koblitz formula to express it in terms of the \(p\)-adic gamma function. He then uses properties of the \(p\)-adic gamma function, binomial coefficients and harmonic sums to reduce the series to the required result.
The author notes that the result does not hold modulo \(p^4\) and that it is an extension of the Apéry number supercongruence, conjectured by F. Beukers [J. Number Theory 25, 201–210 (1987; Zbl 0614.10011)] and proved by S. Ahlgren and K. Ono [J. Reine Angew. Math. 518, 187–212 (2000; Zbl 0940.33002)].


11F33 Congruences for modular and \(p\)-adic modular forms
11F23 Relations with algebraic geometry and topology
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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