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Calabi-Yau manifolds are higher dimensional generalizations of elliptic curves and K3 surfaces. They can be defined as the compact complex Kähler manifolds with trivial canonical bundle, and play a central role in mirror symmetry. This tag can also be used for Calabi-Yau algebras and categories. These algebraic notions are inspired by the properties of the derived categories of coherent sheaves on Calabi-Yau manifolds.
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weight 4 eigenforms with rational coefficients---is it reasonable to expect they all come fr...
I am not that sure that your question can be answered positively, but the following is merely speculation, so you should not pin me down on it. The basic idea is that there might be "more" weight 4 fo …