Worked solution: Potential automorphy of symmetric powers proves the Sato–Tate conjecture (2011)

Step 4 of 8: Automorphy lifting theorems for higher-rank Galois representations

In plain words

Wiles's proof of Fermat's Last Theorem worked by showing that if a 'shadow' of a Galois representation (its reduction modulo a small number) is already known to come from a modular form, then the full representation must also come from one, essentially because there is no room for it to be anything else.

Clozel, Harris, and Taylor extended this 'lifting' technique from the two-dimensional setting of elliptic curves to Galois representations of any dimension, giving a general tool for proving automorphy on $\mathrm{GL}(n+1)$ rather than just $\mathrm{GL}(2)$ .

\bar\rho \text{ automorphic}, \ \rho \text{ a lift of } \bar\rho \ \Longrightarrow \ \rho \text{ automorphic} \qquad \text{(automorphy lifting, CHT 2008)}

Detailed analysis

The Taylor–Wiles method proves an $\ell$ -adic Galois representation $\rho$ is automorphic by comparing a deformation ring (parametrizing all lifts of a fixed residual representation $\bar\rho = \rho \bmod \ell$ ) with a Hecke algebra acting on automorphic forms, an $R = \mathbb{T}$ theorem; when the residual representation $\bar\rho$ is already known to be automorphic and has 'large image', the method shows every suitable lift $\rho$ is automorphic too.

Clozel, Harris, and Taylor (Publ. Math. IHÉS 108, 2008) generalized this Taylor–Wiles–Kisin method from $\mathrm{GL}(2)$ to $n$ -dimensional Galois representations, working with automorphic forms on unitary groups that transfer to $\mathrm{GL}(n)$ via base change; this automorphy lifting theorem is the essential technical engine behind every later step.

The remaining obstacle is that one first needs a residual representation $\bar\rho$ already known to be automorphic to start the lifting process — and for an arbitrary elliptic curve $E$ , no such starting point is obviously available for $\mathrm{Sym}^n E$ . Producing one is the job of the next step.

Terms in this step

Taylor–Wiles method: The technique, invented by Wiles and Taylor for Fermat's Last Theorem, of proving a Galois representation automorphic by comparing a deformation ring with a Hecke algebra (an $R = \mathbb{T}$ theorem), then extended by Clozel, Harris, and Taylor to higher-dimensional representations.
residual representation $\bar\rho$: The reduction of an $\ell$ -adic Galois representation $\rho$ modulo $\ell$ , giving a representation over a finite field; automorphy lifting theorems require this 'shadow' representation to already be known automorphic before lifting to $\rho$ itself.

Knowledge used in this step

Representation theory