We extend the methods of Wiles and of Taylor and Wiles from GL2 to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), minimally ramified, l-adic lifts of certain automorphic mod l Galois representations of any dimension. We also make a conjecture about the structure of mod l automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for GL2. Following Wiles’ method we show that this...

Let Γ ⊂ ℚ * be a finitely generated subgroup and let p be a prime such that the reduction group Γₚ is a well defined subgroup of the multiplicative group ₚ*. We prove an asymptotic formula for the average of the number of primes p ≤ x for which [ₚ*:Γₚ] = m. The average is taken over all finitely generated subgroups $\Gamma =⟨a₁,...,a_r⟩\subset \mathbb{Q}*$, with $a_i\in \mathbb{Z}$ and $a_i\le T_i$, with a range of uniformity $T_i>exp\left(4{\left(logxloglogx\right)}^{1/2}\right)$ for every i = 1,...,r. We also prove an asymptotic formula for the mean square of the error terms in the asymptotic formula with a similar...