Modular Forms and Calabi-Yau Varieties

Abstract

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 $\tau$ acting by $-1$ on $H^0(X, \Omega^{k-1})$. We also raise a general question regarding the regular algebraic, (essentially) selfdual cusp forms $\pi$ on GL$(n)$ with ${\mathbb Q}$-coefficients, asking for associated Calabi-Yau varieties $X=X_\pi$ (with an involution $\tau$ on each such $X$ such that the quotient variety $X/\tau$ is rational) carrying the (conjectural) motive of $\pi$. We then investigate the compatibility of this with Rankin-Selberg products of modular forms.