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

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

2011-11-22

arxiv.org/abs/1111.5275v1

Mathematics

Algebraic Geometry

Scientific paper

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 $\QQ$.

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