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 &#34;ordinarity at 3&#34; easy to check condition, with Hodge-Tate weights $\{0, w \}$ such that $2 w < \ell$ (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 &#34;ordinarity at 3&#34; 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.