Abstract

In a recent paper by F. Gouvea and N. Yui, 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 $$ \mathbb{Q} $$ follows from Serre’s modularity conjecture (now a theorem). In this note, we give an alternative proof of this implication. The main difference with Serre’s argument is that instead of using as a main input residual modularity in infinitely many characteristics, we just require residual modularity in a suitable characteristic. This is combined with the effective Chebotarev theorem.

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