Modularity of Some Non–rigid Double Octic Calabi–yau Threefolds

The modularity conjecture for Calabi–Yau manifolds predicts that every Calabi–Yau manifold should be modular in the sense that its L– series coincides with the L–series of some automorphic form(s). The case of rigid Calabi–Yau threefolds was (almost) solved by Dieulefait and Manoharmayum in [7, 6]. On the other hand in the non–rigid case it is even not clear which automorphic forms should appear. Examples of non–rigid modular Calabi–Yau threefolds were constructed by Livné and Yui ([11]), Hulek and Verrill ([9, 10]) and Schütt ([17]). In these examples modularity means a decomposition of the associated Galois representation into two– and four–dimensional subrepresentations with L–series equal to L(g4, s), L(g2, s−1) or L(g2⊗g3, s), where gk is a weight k cusp form. The summand with L–series equal to L(g2 ⊗ g3, s) is explained by a double cover of a product of a K3 surface and an elliptic curve (see [11]). The L–series L(g2, s − 1) is the L–series of the product of the projective line P and an elliptic curve E with L(E, s) = L(g2, s). A two–dimensional subrepresentation with such an L–series may be identified by a map P × E −→ X which induces a non–zero map on the third cohomology (see [9]). Using an interpretation in terms of deformation theory we conjecture that a splitting of the Galois action into two–dimensional pieces can happen only for isolated elements of any family of Calabi–Yau threefolds. In this paper we will study modularity of some non–rigid double octic Calabi–Yau threefolds, we will prove modularity of all examples listed in table 1 except X154. We will use four methods for proving modularity, apart from the methods of Livné–Yui and Hulek–Verrill we will use two others based on giving a correspondence with a rigid Calabi–Yau