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Modular Forms and Calabi-Yau Varieties

Given a holomorphic newform $f$ of weight $k$ and with rational coefficients, a question of Mazur and van Straten asks if there is an associated Calabi-Yau variety $X$ over ${\mathbb Q}$ of dimension $k-1$ such that the $\ell$-adic Galois representation of $f$ occurs in the cohomology of $X$ in degree $k-1$. We provide some explicit examples giving a positive answer, and show moreover that such $X$ come equipped with an involution $τ$ acting by $-1$ on $H^0(X, Ω^{k-1})$. We also raise a general question regarding the regular algebraic, (essentially) selfdual cusp forms $π$ on GL$(n)$ with ${\mathbb Q}$-coefficients, asking for associated Calabi-Yau varieties $X=X_π$ (with an involution $τ$ on each such $X$ such that the quotient variety $X/τ$ is rational) carrying the (conjectural) motive of $π$. We then investigate the compatibility of this with Rankin-Selberg products of modular forms.