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

By Step 6, for every non-CM $E/\mathbb{Q}$ and every $n \ge 1$, the representation $\mathrm{Sym}^n E$ restricted to some totally real field $F$ is automorphic, so $L(\mathrm{Sym}^n E, s)$ over $F$ enjoys the holomorphic continuation and non-vanishing on $\mathrm{Re}(s) = 1$ guaranteed by Step 3's general automorphic theory.

A standard argument (automorphic induction, or comparing Artin-type factorizations of Dedekind zeta functions of the tower of fields between $\mathbb{Q}$ and $F$) shows the $L$-function over $\mathbb{Q}$ divides the one over $F$ in a suitable sense, and — using that the quotient is itself a product of automorphic $L$-functions with known non-vanishing on $\mathrm{Re}(s) = 1$ — the same holomorphic continuation and non-vanishing descend to $L(\mathrm{Sym}^n E, s)$ over $\mathbb{Q}$ itself, for every $n \ge 1$.

This is precisely the analytic hypothesis Serre's criterion (Step 2) required, now established unconditionally for every non-CM elliptic curve $E/\mathbb{Q}$.