articles of Saito and Yui \[SY\] and of Verrill in Yui \[Y\], these forms were explicitly determined. Saito and Yui use geometric structures; while Verrill uses point counting method, to obtain the results.

More precisely, the following was proved for the natural models over ${{\mathbb Q}}$ of these six rigid threefolds:

[**Theorem 2**]{} (Saito and Yui \[SY\] and Verrill in Yui \[Y\]): [*For each of the six rigid Calabi–Yau threefold over ${{\mathbb Q}}$, the L-series of the third cohomology coincides with the L-series arising from the cusp form of weight $4$ of one variable on the modular group in the Beauville’s list. Beauville’s list and the corresponding cusp forms are given in Table 1.*]{} $$\begin{array}{|c|c|c|} \hline \hline
\mbox{Group} & \mbox{Number of components} &
\mbox{Cusp forms}    \\
\Gamma       & \mbox{of singular fibers}   &
\mbox{of weight $4$}  \\ \hline
\Gamma(3)   & 3,3,3,3                &
\eta(q^3)^8   \\ \hline
\Gamma_1(4)\cap \Gamma(2) & 4,4,2,2  &
\eta(q^2)^4\eta(q^4)^4 \\ \hline
\Gamma_1(5) & 5,5,1,1                &
\eta(q)^4\eta(q^5)^4   \\ \hline
\Gamma_1(6) & 6,3,2,1              &
\eta(q)^2\eta(q^2)^2\eta(q^3)^2\eta(q^6)^2 \\
\hline
\Gamma_0(8)\cap\Gamma_1(4) & 8,2,1,1&
\eta(q^4)^{16}\eta(q^2)^{-4}\eta(q^8)^{-4} \\
\hline
\Gamma_0(9)\cap\Gamma_1(3) & 9,1,1,1 &
\eta(q^3)^8      \\ \hline \hline
\end{array}$$

Table 1: Rigid Calabi–Yau threefolds and cusp forms

Here $\eta(q)$ denotes the Dedekind eta-function: $\eta(q)=q^{1/24}\prod_{n\geq 1}(1-q^n)$ with $q=e^{2\pi i\tau}$.

