Are periods of rigid Calabi-Yau threefolds over $Q$ algebraic? - MathOverflow

Are periods of rigid Calabi-Yau threefolds over $Q$ algebraic?

2010-06-13T21:22:16Z

Let $X$ be a (smooth) compact complex manifold, and suppose that $H^1(X, \Theta_X) = 0$, where $\Theta_X$ is the tangent sheaf. In other words, suppose that $X$ is rigid.

Suppose moreover that $X$ arises as the complex points of a smooth projective variety over $Q$.

Is it known or expected that the periods of $X$ are algebraic numbers? If $X$ were not rigid, then the periods would be values (at zero) of functions satisfying Picard-Fuchs equations. But the rigidity suggests (to my intuition) that the periods should not be transcendental.

Is anything known? Expected? Written? How about the specific case when $X$ is a rigid Calabi-Yau 3-fold? Has anyone computed such periods? Could one compute them easily?

Answer by Junkie for Are periods of rigid Calabi-Yau threefolds over $Q$ algebraic?

2010-06-14T01:29:04Z

For rigid, at least in the modular case (known in many events), you can compute the periods of the form, though this supposes you can explicitly write down the weight 4 newform. For instance, Schutt ( http://arxiv.org/pdf/math/0311106 ) gives examples of level 73, and using Magma you can compute the periods as

> M:=NewformDecomposition(NewSubspace(CuspidalSubspace(ModularSymbols(73,4))))[1];
> Periods(M,100);
[ (0.902834199842382836695960181248 + 0.0526923557275574794028757363126*i),
  (0.285105536792331422114513708795 + 0.0175641185758524931342918404798*i) ]

Here $L$-functions are not applicable, as the $L$-functions vanishes at the central point. I extended the above to a few hundred digits and found nothing with PowerRelation.