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

2004/09/07 by Luis Dieulefait, Luís Dieulefait, Dieulefait, Luis
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #math.NT

paper · pdf · doi:10.48550/arxiv.math/0409102

arxiv created 2004/09/07 · arxiv updated 2009/12/01

Abstract

We prove modularity for any irreducible crystalline ℓ-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 < ℓ (and ℓ > 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(\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 a₃ are modular.

Citations

• On ordinary ?-adic representations associated to modular forms

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