Degeneration of Calabi-Yau threefolds with Galois action induced by modular form of weight 3

Abstract

We study a one-parameter family of double octic Calabi-Yau threefolds with two degenerations of type $K$. For each of these degenerations we construct a semistable model and compute the limiting mixed Hodge structure. We show that the $\ell$-adic Galois representation on the limiting cohomology is isomorphic (up to semisimplification) to the direct sum of the Galois representation attached to a weight-three modular form associated with a singular K3 surface and its Tate twist. This gives an explicit realization of the expected relation between type $K$ degenerations of Calabi-Yau threefolds and K3 surfaces.

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