Gouvea and Yui on modularity of rigid CY 3-folds.
arXiv:0908.1210n the modularity of rigid Calabi-Yau threefolds: Epilogue from arXiv Front: math.NT by Luis Dieulefait 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.
See a chapter in Global aspects of Complex Geometry, in Complex manifolds or Conferences folder
nLab page on Modularity