Hilbert modularity of some double octic Calabi--Yau threefolds

TL;DR

This paper demonstrates the modularity of three double octic Calabi--Yau threefolds over quadratic fields, linking them to specific Hilbert modular forms and expanding understanding of their arithmetic properties.

Contribution

It provides explicit examples of double octic Calabi--Yau threefolds over quadratic fields and establishes their modularity with detailed correspondence to Hilbert modular forms.

Findings

Non-rigid threefold linked to conjugate Hilbert modular forms of weights [4,2] and [2,4]

Two rigid threefolds associated with a Hilbert modular form of weight [4,4] and a twisted classical modular form

Explicit modularity proofs for these Calabi--Yau threefolds over quadratic fields

Abstract

We exhibit three double octic Calabi--Yau threefolds over the certain quadratic fields and prove their modularity. The non-rigid threefold has two conjugate Hilbert modular forms of weight [4,2] and [2,4] attached while the two rigid threefolds correspond to a Hilbert modular form of weight [4,4] and to the twist of the restriction of a classical modular form of weight 4.