From potential modularity to modularity for integral Galois representations and rigid Calabi-Yau threefolds

Abstract

We prove modularity for any irreducible crystalline $\ell$-adic odd 2-dimensional Galois representation (with finite ramification set) unramified at 3 verifying an "ordinarity at 3" easy to check condition, with Hodge-Tate weights $\{0,w\}$ such that $2w<\ell$ (and $\ell >3$) and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p \}) = \Q $. This result applies in particular to any motivic compatible family of odd two-dimensional Galois representations of $\Gal(\bar{\Q}/\Q)$ if the motive has rational coefficients, good reduction at 3, and the "ordinarity at 3" condition is satisfied. As a corollary, this proves that all rigid Calabi-Yau threefolds defined over $\Q$ having good reduction at 3 and satisfying $3\nmid a_3$ are modular.

Cited by in corpus (1)

• On the modularity of Calabi-Yau threefolds containing elliptic ruled surfaces