On the modularity of rigid Calabi-Yau threefolds: Epilogue

TL;DR

This paper provides an alternative proof that all rigid Calabi-Yau threefolds over rationals are modular, relying on residual modularity in a specific characteristic and effective Cebotarev, simplifying previous patching arguments.

Contribution

It introduces a new proof method for modularity of rigid Calabi-Yau threefolds, reducing the reliance on residual modularity in infinitely many characteristics.

Findings

Alternative proof of modularity for rigid Calabi-Yau threefolds

Reduction of assumptions needed for modularity proof

Application of effective Cebotarev in the proof

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.