TITLE: weight 4 eigenforms with rational coefficients---is it reasonable to expect they all come from Calabi-Yaus?
QUESTION [19 upvotes]: A weight 2 modular form which happens to be a normalised cuspidal eigenform with rational coefficients has a natural geometric avatar---namely an elliptic curve over the rationals. It seems to be a subtle question as to how this notion generalises---this question was raised by Tony Scholl in conversation with me the other day. For example I guess I wouldn't expect a weight 3 normalised cuspidal eigenform with rational coefficients to be the $H^2$ of a smooth projective surface, because any such surface worth its salt would have (1,1)-forms coming from a hyperplane section, whereas the Hodge numbers of the motive attached to a weight 3 form are 0 and 2.
But in weight 4 one can again dream. A rigid Calabi-Yau 3-fold defined over the rationals has 2-dimensional $H^3$ and the Hodge numbers match up. Indeed there are many explicit examples of pairs $(X,f)$ with $X$ a rigid Calabi-Yau 3-fold over $\mathbf{Q}$ and $f$ a weight 4 cuspidal modular eigenform, such that the $\ell$-adic Galois representation attached to $f$ is isomorphic to $H^3(X,\mathbf{Q}_\ell)$ for all $\ell$.

The question: Is it reasonable to expect that (the motive attached to) every weight 4 normalised cuspidal eigenform with rational coefficients is associated to the cohomology of a rigid Calabi-Yau 3-fold over $\mathbf{Q}$?

REPLY [3 votes]: To follow up the question of the intermediate Jacobian, there is indeed a later survey by Noriko Yui (Arithmetic of Calabi-Yau varieties.  Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Summer Term 2004,  9--29) where she does conjecture it to be defined over $\mathbb Q$ for any (modular) rigid CY3 over $\mathbb Q$. 
Moreover, she refers to a joint paper with X. Xarles in preparation (still unpublished) which claims to settle this in the CM case:
Let $X$ be a rigid CY3 of CM type (i.e. with commutative Hodge group) over some number field $F$. Then the intermediate jacobian $J^2(X)$ is an elliptic curve with CM by an order in an imaginary quadratic field (understood: the same field, since there will be a relation of Hecke characters over some extension), and it has a model over $F$.
Let's apply this to rigid CY3's over $\mathbb Q$ and assume that there is one for each newform of weight 4 with rational coefficients. Pick one of the weight 4 newforms with rational coefficients and CM of class number 3, i.e. induced by a Hecke character for an imaginary quadratic field $K$ of class number 3 such as $\mathbb Q(\sqrt{-23})$ (or generally of class group exponent 3). By assumption there is an associated CY3 $X$ over $\mathbb Q$ (which ought to have CM). But then, by the above result, its intermediate Jacobian is an elliptic curve over $\mathbb Q$ with CM in $K$, contradiction.
Of course, one can still ask whether all non-CM newforms can be realized in some CY3's over $\mathbb Q$, but I would find it surprising if this would only fail at certain CM forms. And after all, I would be glad to allow non-rigid CY3's over $\mathbb Q$ admitting the right submotive, but this might still not be sufficient.