| Résume | We will discuss the modularity of Galois representations associated toCalabi--Yau threefolds over $\mathbf Q$. We can show that any rigid Calabi--Yau threefold over $\mathbf Q$ is modulari.e., its two-dimensional Galois representation comes from a modular form of weight $4$ on some some congruence subgroup of $PSL_2(\mathbf Z)$. However, when a Calabi--Yauthreefold is non-rigid, the dimension of the Galois representation getsrather large, and the modularity question poses a serious challenge.We will construct explicit examples of non-rigidCalabi--Yau threefolds fibered over ${\bf P}^1$ by non-constantsemi-stable $K3$ surfaces and reaching the Arakelov--Yauupper bound. For these examples, we prove that the ``interesting'' partof their $L$-series do come from modular forms. |
