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

Step 6 of 8: Assembling the full result: Taylor (2008) and Barnet-Lamb–Geraghty–Harris–Taylor (2011)

In plain words

Richard Taylor, refining the potential-automorphy machinery from Step 5, proved in 2008 that $\mathrm{Sym}^n E$ is potentially automorphic for every $n$ , but only for elliptic curves satisfying an extra technical hypothesis (having 'multiplicative reduction' — a mild singularity — at some prime).

Three years later, Barnet-Lamb, Geraghty, Harris, and Taylor built a better family of Calabi–Yau varieties and improved automorphy lifting theorems that removed this hypothesis entirely, extending potential automorphy of $\mathrm{Sym}^n E$ to every non-CM elliptic curve over $\mathbb{Q}$ .

\mathrm{Sym}^n E \text{ potentially automorphic for every non-CM } E/\mathbb{Q} \text{ and every } n \ge 1 \qquad \text{(BGHT, 2011)}

Detailed analysis

Taylor (2008, two papers in Publ. Math. IHÉS and Annals of Math extending CHT) proved that for a non-CM elliptic curve $E/\mathbb{Q}$ with multiplicative reduction at some prime, $\mathrm{Sym}^n E$ is potentially automorphic for every $n \ge 1$ ; combined with Steps 2–3, this already gave the Sato–Tate conjecture for this restricted (but common) class of curves — the first proof of any case of the conjecture.

Barnet-Lamb, Geraghty, Harris, and Taylor (2011, Publ. Math. RIMS, 'A family of Calabi–Yau varieties and potential automorphy II') then constructed an improved family of Calabi–Yau varieties robust enough to supply the needed residual automorphy in all cases, and refined the automorphy lifting theorems to remove the multiplicative-reduction hypothesis entirely, proving potential automorphy of $\mathrm{Sym}^n E$ for every non-CM elliptic curve $E/\mathbb{Q}$ and every $n \ge 1$ .

With potential automorphy now unconditional, all that remains is to transfer the resulting analytic properties from the auxiliary field $F$ back down to $\mathbb{Q}$ , closing the loop opened in Step 3.

Knowledge used in this step

Arithmetic geometry