Modularity of some non-rigid double octic Calabi-Yau threefolds

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In this paper we discuss four methods of proving modularity of Calabi--Yau threefolds with h¹²=1: existence of elliptic ruled surfaces inside (Hulek-Verrill), correspondence with a product of an elliptic curve and a K3 surface (Livné-Yui), correspondence with a (modular) rigid Calabi-Yau threefold, and existence of an involution splitting the fourdimensional representation into twodimensional subrepresentations. We apply these methods to prove modularity of 17 out of 18 double octic Calabi-Yau threefolds for which "numerical evidence of modularity" was found in the second Author's thesis. We observe that modularity holds for those elements in a pencil having some additional geometric properties. In the proofs we use representations of the considered Calabi-Yau threefolds as a Kummer fibration associated to a fiber product of rational elliptic fibrations.