Modularity conjecture for rigid Calabi–Yau threefolds

Recall that a smooth, projective threefold $X$ is a Calabi–Yau threefold if its canonical bundle is trivial and $H^{1}(X,\mathcal{O}_{X})=H^{2}(X,\mathcal{O}_{X})=0$. It is rigid if $H^{1}(X,\mathcal{T}_{X})=0$. For a rigid Calabi–Yau threefold defined over $\mathbb{Q}$, let $L(X,s)$ denote its $L$-series.

Modularity conjecture. Any rigid Calabi–Yau threefold $X$ defined over $\mathbb{Q}$ is modular: its $L$-series $L(X,s)$ coincides up to finitely many Euler factors with the Mellin transform $L(f,s)$ of a modular cusp form $f$ of weight $4$ with respect to $\Gamma_{0}(N)$, where the level $N$ is only divisible by the primes of bad reduction.

This conjecture predicts that arithmetic data from rigid Calabi–Yau threefolds over $\mathbb{Q}$ are governed by weight-$4$ modular forms. The paper constructs examples realizing some new weight-$4$ cusp forms, while the general modularity assertion remains open.