Answer by Matthias Schuett for weight 4 eigenforms with rational coefficients---is it reasonable to expect they all come from Calabi-Yaus?

2010-09-03T14:16:21Z

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Moreover, she refers to a joint paper with X. Xarles in preparation (still unpublished) which claims to settle this in the CM case: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.

User matthias schuett - MathOverflow

Answer by Matthias Schuett for weight 4 eigenforms with rational coefficients---is it reasonable to expect they all come from Calabi-Yaus?