Descent of a residual representation to Gal(L₀/ℚ)

Let $k$ be a field and let $\bar\rho$ be a residual Galois representation over $k$: a $k$-vector space $V$ with $\dim_k V = 2$, a monoid homomorphism $\bar\rho\colon \operatorname{Aut}_{\mathbb{Q}}(\overline{\mathbb{Q}}) \to \operatorname{End}_k V$ (where $\overline{\mathbb{Q}}$ is AlgebraicClosure ℚ), together with the datum of some intermediate field $L \subseteq \overline{\mathbb{Q}}$, finite over $\mathbb{Q}$, such that $\bar\rho(\sigma) = 1$ whenever $\sigma$ fixes $L$ pointwise. Let $L_0 \subseteq \overline{\mathbb{Q}}$ be an intermediate field which is finite-dimensional over $\mathbb{Q}$, a number field, and Galois over $\mathbb{Q}$, and assume $\bar\rho(\sigma) = 1$ for every $\sigma \in \operatorname{Aut}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ fixing $L_0$ pointwise. Let $b$ be a basis of $V$ indexed by Fin 2. Then there exists a monoid homomorphism $\rho_{\mathrm{mat}}$ from $\operatorname{Aut}_{\mathbb{Q}}(L_0)$ to the multiplicative monoid of $2 \times 2$ matrices over $k$ such that for every $\sigma \in \operatorname{Aut}_{\mathbb{Q}}(\overline{\mathbb{Q}})$, the value of $\rho_{\mathrm{mat}}$ at the restriction of $\sigma$ to $L_0$ equals the matrix of $\bar\rho(\sigma)$ in the basis $b$. The descent depends on $b$; the proof uses only the normality of $L_0$ over $\mathbb{Q}$, not the finiteness or number-field hypotheses, nor the finite level attached to $\bar\rho$.

This is the standard descent step of Galois theory: a homomorphism out of $\operatorname{Aut}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ that is trivial on the automorphisms fixing a normal subextension $L_0$ factors through the restriction map onto $\operatorname{Aut}_{\mathbb{Q}}(L_0)$. It converts a residual Galois representation into a matrix representation of a finite Galois group, the form in which the Taylor–Wiles prime arguments operate; it is used in the production of Taylor–Wiles primes avoiding a given set for absolutely irreducible residual representations.