On the modularity of rigid Calabi-Yau threefolds: Epilogue

Abstract

In a recent preprint of F. Gouvea and N. Yui (see arXiv:0902.1466) a detailed account is given of a patching argument due to Serre that proves that the modularity of all rigid Calabi-Yau threefolds defined over the rationals follows from Serre's modularity conjecture. In this note (a letter to N. Yui) we give an alternative proof of this implication. The main difference with Serre's argument is that instead of using as main input residual modularity in infinitely many characteristics we just require residual modularity in a suitable characteristic. This is combined with effective Cebotarev.

explanations expanded. To appear in "Proceedings of the trimester on Diophantine Equations at the Hausdorff Institute", Zapiski POMI of the Steklov Math. Inst. St. Petersbourg (and J. Math. Sciences)