Periods of singular double octic Calabi-Yau threefolds and modular forms

Tymoteusz Chmiel; Sławomir Cynk

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arXiv · 2202.01556

Periods of singular double octic Calabi-Yau threefolds and modular forms

Tymoteusz Chmiel·Sławomir Cynk

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Abstract

By the modularity theorem every rigid Calabi-Yau threefold $X$ has associated modular form $f$ such that the equality of $L$-functions $L(X,s)=L(f,s)$ holds. In this case period integrals of $X$ are expected to be expressible in terms of the special values $L(f,1)$ and $L(f,2)$. We propose a similar interpretation of period integrals of a nodal model of $X$. It is given in terms of certain variants of a Mellin transform of $f$. We provide numerical evidence towards this interpretation based on a case of double octics.

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Tymoteusz Chmiel, Sławomir Cynk. 2022-02-03. Periods of singular double octic Calabi-Yau threefolds and modular forms. https://arxiv.org/abs/2202.01556

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