Lifting N-dimensional Galois representations to characteristic zero

Let F be a number field, let N ≥ 3 be an integer, and let k be a finite field of characteristic ℓ. We show that if ρ:GF → GLN(k) is a continuous representation with image of ρ containing SLN(k) then, under moderate conditions at primes dividing ℓ∞, there is a continuous representation ρ:GF → GLN(W(k)) unramified outside finitely many primes with ρ ~ρ mod ℓ. Stronger results are presented for ρ:Gℚ → GL3(k)

A structure theorem for subgroups of GLn over complete local Noetherian rings with large residual image

Given a complete local Noetherian ring (A, mA) with finite residue field and a subfield k of A/mA, we show that every closed subgroup G of GLn(A) such that G mod mA ⊇ SLn(k) contains a conjugate of SLn(W(k)A) under some small restrictions on k. Here W(k)A is the closed subring of A generated by the Teichm¨uller lifts of elements of the subfield k

Elliptic Curves over Real Quadratic Fields are Modular

We prove that all elliptic curves defined over real quadratic fields are modular.Comment: 38 pages. Magma scripts available as ancillary files with this arXiv versio

A note on the structure of complete alternative local algebras

Let (A, m) be an alternative algebra with maximal ideal m which is complete and separated for the m-adic topology. Assuming that A/m := k is a perfect field of positive characteristic and that the associated graded algebra is a k-algebra, we show that the reduction map W(k) → k from the Witt ring W(k) lifts canonically to a morphism W(k) → A thereby giving A the structure of a unital W(k)-algebra

Modularity of rigid Calabi-Yau threefolds over Q

We prove modularity for a huge class of rigid Calabi-Yau threefolds over Q. In particular we prove that every rigid Calabi-Yau threefold with good reduction at 3 and 7 is modular