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

Step 5 of 8: Potential automorphy: borrow a residual representation from geometry

In plain words

Since there is no ready-made automorphic residual representation to start the lifting machine directly over $\mathbb{Q}$ , Harris, Shepherd-Barron, and Taylor instead searched for $\mathrm{Sym}^n E$ hiding inside the cohomology of an auxiliary family of higher-dimensional shapes called Calabi–Yau varieties, chosen so that at special members of the family the residual representation is already known automorphic (often by explicit, classical methods).

Applying the lifting theorem then only requires passing to a suitable auxiliary number field $F$ where the geometric coincidence lines up — giving automorphy of $\mathrm{Sym}^n E$ not over $\mathbb{Q}$ itself, but 'potentially', after this base change.

\mathrm{Sym}^n E \hookrightarrow H^*(\text{Dwork family}) \ \Longrightarrow \ \mathrm{Sym}^n E|_F \text{ automorphic for some totally real } F

Detailed analysis

Harris, Shepherd-Barron, and Taylor realized $\mathrm{Sym}^n E$ inside the middle cohomology of a specific family of Calabi–Yau varieties (built from Dwork-type hypersurfaces), chosen precisely so that at certain fibers the corresponding residual Galois representation is automorphic by classical results, and so that the family's geometry supplies the 'big image' condition the lifting theorem needs.

Because the required coincidence can generally only be arranged after enlarging the base field, the automorphy lifting theorem yields automorphy of $\mathrm{Sym}^n E$ restricted to $\mathrm{Gal}(\overline{\mathbb{Q}}/F)$ for some auxiliary totally real (or CM) number field $F$ , rather than over $\mathbb{Q}$ itself — a weaker conclusion called potential automorphy, introduced in this form by Harris, Shepherd-Barron, and Taylor (2010, 'Algebraic families of Galois representations and potentially automorphic representations').

This potential-automorphy strategy is the central innovation that makes the whole Sato–Tate proof possible: it sidesteps the missing starting point over $\mathbb{Q}$ by manufacturing one geometrically, at the cost of first proving automorphy only after a base change.

Terms in this step

potential automorphy: The property that a Galois representation becomes automorphic only after restricting it to (base-changing to) some finite extension field $F$ of $\mathbb{Q}$ , rather than being automorphic over $\mathbb{Q}$ itself; a weaker but often sufficient substitute when direct automorphy cannot yet be proved.
Calabi–Yau variety: A special class of higher-dimensional algebraic varieties with a trivial canonical bundle; families of such varieties (like Dwork hypersurfaces) can be engineered so their cohomology realizes prescribed Galois representations, such as $\mathrm{Sym}^n E$ .

Knowledge used in this step

Algebraic geometry Modular forms