Hypergeometric families of Calabi-Yau manifolds. (English) Zbl 1062.11038

Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3355-3/hbk). Fields Inst. Commun. 38, 223-231 (2003).
Let \(\gamma = \sum_{\nu \geq 1} \gamma_\nu [\nu]\) be a hypergeometric weight system (the \(\gamma_\nu \in \mathbb{Z}\) are zero for all but finitely many \(\nu\) and satisfy (i) \(\sum_{\nu\geq 1} \nu \gamma_\nu = 0\) (ii) \(d = d(\nu) = - \sum_{ \nu \geq 1} \gamma_\nu > 0\); the number \(d\) is called the dimension of \(\gamma\)). To any such \(\gamma\) one can associate the hypergeometric function \(u(\lambda) = \sum_{n \geq 0} \lambda^n\) where the \(u_n\)’s are given by \(u_n = \prod_{\nu \geq 1} (\nu^n)!^{ \gamma_\nu}\) and condition (i) ensures that the linear differential equation (of order \(r\)) satisfied by \(u\) has only regular singularities. Here \(r\) is minimal such that \[ u(\lambda) = {}_r F_{r - 1} \left(\left.\begin{matrix} \alpha_1 \ldots \alpha_r\\ \beta_1 \ldots \beta_r \end{matrix} \right| \dfrac{\lambda}{\lambda_o} \right) \] where \(\lambda_0\) is given explicitly.
The author looks for \(\gamma\) as above such that the \(p\)-truncation of \(u\) (\(p\)-prime) is related to the number of points over \(\mathbb{F}_p\) of some family of varieties \(X_\lambda\), and for this one needs that the \(u_n\) are integers for all \(n = 0, 1, 2, \ldots\).
The author proves (using Landau’s function \(\mathcal{L} (x)\)) that if \(\gamma\) is not integral then for all \(p\) sufficiently large there is \(n\), \(0 \leq n < p\) such that \(V_p (u_n) < 0\). The author gives a list of 22 generators of the cone of integral \(\gamma\) with support \(\mathcal{N} = 1,2,3,4,5,6,10,15,30\).
The author announces (the proof, to be published elsewhere) that if \(\gamma\) is a hypergeometric weight system, the associated hypergeometric function is algebraic iff \(\gamma\) is integral of dimension \(d = 1\). He also affirms that for the Chebyshev’s example the monodromy group of the corresponding differential equation is the Weyl group of the \(E_8\) lattice.
Also, the author has found numerically a phenomenon of supercongruence (i.e. mod \(p^3\)), and gives some examples.
For the entire collection see [Zbl 1022.00014].


11G25 Varieties over finite and local fields
11A07 Congruences; primitive roots; residue systems
33C20 Generalized hypergeometric series, \({}_pF_q\)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32Q25 Calabi-Yau theory (complex-analytic aspects)