Rigid realizations of modular forms in Calabi–Yau threefolds

Abstract

We construct examples of modular rigid Calabi-Yau threefolds, which give a realization of some new weight 4 cusp forms. Involutions of double octic Calabi-Yau threefoldsLet S ⊂ P 3 be an arrangement of eight planes such that no six intersect and no four contain a line. Then there exists a resolution of singularities X of the double covering of P 3 branched along S which is a Calabi-Yau threefold.

Counting points on double octic Calabi–Yau threefolds over finite fields

“…not affecting the canonical class) resolution of singularities of singular degenerations of previously known Calabi-Yau threefolds. Usually this construction produces a threefold with a smaller Hodge number h 1,2 . A double cover of a smooth manifold is singular exactly when the branch locus is singular.…”

“…4 fourfold point not lying on a triple line, p 1 4 fourfold point lying on one triple line, p 0 5 fivefold point not lying on a triple line, p 1 5 fivefold point lying on one triple line, p 2 5 fivefold point lying on two triple lines. We denote with the same symbols the number of multiple lines or points of a given type.…”

“…Proof. The projective transformation φ : (x, y, z, t) → (x, z, y, t) preserves the octic arrangement D, so it lifts to an automorphism Φ : X → X of Calabi-Yau threefold (for details see [1]). In fact there are two such liftings that differ by the sheet exchanging involution; we choose the one that preserves the canonical form of X.…”