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The Sato-Tate conjecture for Hilbert modular forms. (English) Zbl 1269.11045
This important and clearly-written article proves the following natural extension of the Sato-Tate conjecture to the setting of cuspidal automorphic representations of $$\text{GL}_2$$ over totally real number fields (see, Theorem 7.1.6 and Corollary 7.1.7).
Theorem. Let $$F$$ be a totally real number field. Let $$\pi = \otimes_v \pi_v$$ be a non-CM regular algebraic cuspidal automorphic representation of $$\text{GL}_2$$ over $$F$$. Let $$\psi$$ be the finite-order character obtained by taking the product of the central character of $$\pi$$ with $$| \cdot |^{w_{\pi}}$$. Let $$\zeta$$ be a root of unity for which $$\zeta^2$$ is contained in the image of $$\psi$$. Given a place $$v$$ of $$F$$ for which the local representation $$\pi_v$$ is unramified, let $$t_v$$ be the eigenvalue of the Hecke operator $\left[ \text{GL}_2(\mathcal{O}_{F_v}) \left(\begin{matrix} \varpi_v & 0 \\ 0 & 1 \end{matrix}\right) \text{GL}_2(\mathcal{O}_{F_v}) \right]$ acting on $$\pi_v^{\text{GL}_2(\mathcal{O}_{F_v})}$$, where $$\varpi_v$$ is a uniformizer of $$\mathcal{O}_{F_v}$$. Note that if $$\psi_v(\varpi_v) = \zeta^2$$, then $$t_v/(2 ({\mathbb{N}}v)^{(1+w_v)/2} \zeta )$$ lies in the interval $$[-1, 1] \subset {\mathbb{R}}$$.
Then, as $$v$$ ranges over places $$F$$ for which $$\pi_v$$ is unramified and $$\psi_v(\varpi_v) = \zeta^2$$, the values $$t_v/(2 ({\mathbb{N}}v)^{(1+w_v)/2} \zeta )$$ are equidistributed in $$[-1, 1]$$ with respect to the measure $$(2/\pi) \sqrt{1-t^2}\,dt$$.
The proof, which uses as main ingredients (i) a method of congruences developed by T. Gee [Doc. Math., J. DMV 14, 771–800 (2009; Zbl 1246.11100)], (ii) a twisting trick due to M. Harris [Prog. Math. 270, 1–21 (2009; Zbl 1234.11068)], (cf. [T. Barnet-Lamb, D. Geraghty, M. Harris and R. Taylor, Publ. Res. Inst. Math. Sci. 47, No. 1, 29–98 (2011; Zbl 1264.11044)]), and (iii) an automorphy lifting theorem shown in $$\S 3$$, is sketched in the introduction. The details are then given in the seventh section, with background results contained in the five intermediate sections.
This result of course builds on the seminal works of M. Harris, N. Shepherd-Barron and R. Taylor [Ann. Math. (2) 171, No. 2, 779–813 (2010; Zbl 1263.11061)] and R. Taylor [Publ. Math., Inst. Hautes Étud. Sci. 108, 183–239 (2008; Zbl 1169.11021)], along with other antecedent results and techniques, as explained nicely in the introduction.

MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F80 Galois representations
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