Modular Calabi–yau Threefolds of Level Eight

If X̃ is a Calabi–Yau threefold defined over Q, and p is a good prime (i.e., a prime such that the reduction of X̃ mod p is nonsingular) then the map Frob∗p : H i ét(X̃,Ql) −→ H i ét(X̃,Ql) on l–adic cohomology induced by the geometric Frobenius morphism gives rise to l–adic Galois representations ρl,i : Gal(Q/Q) −→ GLbi(Ql). If a Calabi–Yau threefold X̃ is rigid (i.e., h(X̃) = 0 or equivalently b(X̃) = 2) then X̃ is expected to be modular, i.e., the L–series of the (semi-simplification of the) two-dimensional Galois representation ρl,3 associated with the middle cohomology H ét(X̃,Ql) equals the L–series of a cusp form f of weight 4 for Γ0(N). The precise conjecture has been formulated by Saito and Yui in [10]. For details and examples the reader is referred to [14] or [6]. There are many examples of pairs of Calabi–Yau threefolds with an isomorphism between some pieces of their middle étale cohomologies and the appropriate Galois representations. In particular, if we can attach modular forms to these pieces then these modular forms will be the same. If on the other hand we detect the same modular forms in the middle étale cohomologies of two Calabi–Yau threefolds then this should have a geometrical reason: