Quadratic twists of rigid Calabi-Yau threefolds over $\QQ$

TL;DR

This paper explores the concept of quadratic twists of rigid Calabi-Yau threefolds over the rationals, linking geometric twists to modular forms and examining which newforms correspond to such threefolds.

Contribution

It provides a precise geometric definition of quadratic twists for rigid Calabi-Yau threefolds over  and relates these twists to quadratic characters of attached modular forms.

Findings

Quadratic twists correspond to twisting the associated newform by quadratic characters.

Examples illustrate both obvious and subtle cases of quadratic twists.

The work connects geometric properties of threefolds with modular form theory.

Abstract

We consider rigid Calabi--Yau threefolds defined over $\QQ$ and the question of whether they admit quadratic twists. We give a precise geometric definition of the notion of a quadratic twists in this setting. Every rigid Calabi--Yau threefold over $\QQ$ is modular so there is attached to it a certain newform of weight 4 on some $\Gamma_0(N)$. We show that quadratic twisting of a threefold corresponds to twisting the attached newform by quadratic characters and illustrate with a number of obvious and not so obvious examples. The question is motivated by the deeper question of which newforms of weight 4 on some $\Gamma_0(N)$ and integral Fourier coefficients arise from rigid Calabi--Yau threefolds defined over $\QQ$.