Classical hypergeometry and the modularity of Calabi–Yau manifolds

I will examine instances of modularity of (rigid) Calabi--Yau manifolds whose periods are expressed in terms of
hypergeometric functions. The $p$-th coefficients $a(p)$ of the corresponding modular form can be often read off,
at least conjecturally, from the truncated partial sums of the underlying hypergeometric series modulo a power of
$p$ and from Weil's general bounds $|a(p)|\le2p^{(m-1)/2}$, where $m$ is the weight of the form. Furthermore,
the critical $L$-values of the modular form are predicted to be rationally proportional to the values of a related
basis of solutions to the hypergeometric differential equation.