# zbMATH — the first resource for mathematics

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].

##### MSC:
 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)