This paper investigates the modularity of three non-rigid Calabi--Yau threefolds with bad reduction at 11. They are constructed as fibre products of rational elliptic surfaces, involving the modular elliptic surface of level 5. Their middle $\ell$-adic cohomology groups are shown to split into two-dimensional pieces, all but one of which can be interpreted in terms of elliptic curves. The remaining pieces are associated to newforms of weight 4 and level 22 or 55, respectively. For this purpose, we develop a method by Serre to compare the corresponding two-dimensional 2-adic Galois representations with uneven trace. Eventually this method is also applied to a self fibre product of the Hesse-pencil, relating it to a newform of weight 4 and level 27.

MSC Classifications:

14J32, 11F11, 11F23, 20C12 show english descriptions Calabi-Yau manifolds
Holomorphic modular forms of integral weight
Relations with algebraic geometry and topology
Integral representations of infinite groups 14J32 - Calabi-Yau manifolds
11F11 - Holomorphic modular forms of integral weight
11F23 - Relations with algebraic geometry and topology
20C12 - Integral representations of infinite groups

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