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

TL;DR

This paper proves modularity for certain 2-dimensional Galois representations with specific properties, leading to the modularity of rigid Calabi-Yau threefolds over rationals under particular conditions.

Contribution

It establishes modularity for a broad class of Galois representations and applies this to prove the modularity of specific rigid Calabi-Yau threefolds over Q.

Findings

Proves modularity for irreducible crystalline Galois representations under new conditions.

Shows all rigid Calabi-Yau threefolds over Q with good reduction at 3 and specific trace conditions are modular.

Extends the potential modularity results to integral Galois representations with explicit criteria.

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 $2 w < \ell$ (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 "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.