The modularity conjecture for rigid Calabi–Yau threefolds over Q

Let $X$ be a rigid Calabi–Yau threefold defined over ${\mathbb Q}$ with a suitable integral model. Its $L$-series is the $L$-series of the Galois representation on $H^3_{\operatorname{et}}(\bar{X},{\mathbb Q}_{\ell})$. Let $S_4(\Gamma_0(N))$ denote the space of cusp forms of weight $4$ for $\Gamma_0(N)$.

The modularity conjecture. Any rigid Calabi–Yau threefold $X$ defined over ${\mathbb Q}$ is modular: its $L$-series coincides, up to finitely many Euler factors, with the $L$-series of a cusp form $f$ of weight $4$ on $\Gamma_0(N)$, where $N$ is a positive integer divisible by the primes of bad reduction. More precisely,

$$L(X,s)=L(f,s),\qquad f\in S_4(\Gamma_0(N)).$$

This is intended as a dimension-three analogue of the Taniyama–Shimura–Weil modularity conjecture for elliptic curves. The elliptic-curve conjecture has been proved, whereas the asserted modularity of rigid Calabi–Yau threefolds is the subject of the paper and is not established in general.