Are periods of rigid Calabi-Yau threefolds over \$Q\$ algebraic? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T10:56:07Z http://mathoverflow.net/feeds/question/28066 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28066/are-periods-of-rigid-calabi-yau-threefolds-over-q-algebraic Are periods of rigid Calabi-Yau threefolds over \$Q\$ algebraic? Marty 2010-06-13T21:22:16Z 2010-06-14T01:29:04Z <p>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.</p> <p>Suppose moreover that \$X\$ arises as the complex points of a smooth projective variety over \$Q\$.</p> <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/28066/are-periods-of-rigid-calabi-yau-threefolds-over-q-algebraic/28078#28078 Answer by Junkie for Are periods of rigid Calabi-Yau threefolds over \$Q\$ algebraic? Junkie 2010-06-14T01:29:04Z 2010-06-14T01:29:04Z <p>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 ( <a href="http://arxiv.org/pdf/math/0311106" rel="nofollow">http://arxiv.org/pdf/math/0311106</a> ) gives examples of level 73, and using Magma you can compute the periods as</p> <pre><code>&gt; M:=NewformDecomposition(NewSubspace(CuspidalSubspace(ModularSymbols(73,4))))[1]; &gt; Periods(M,100); [ (0.902834199842382836695960181248 + 0.0526923557275574794028757363126*i), (0.285105536792331422114513708795 + 0.0175641185758524931342918404798*i) ] </code></pre> <p>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.</p>