×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. · Zbl 0682.10022
[2] Emil Artin and John Tate, Class field theory, AMS Chelsea Publishing, Providence, RI, 2009. Reprinted with corrections from the 1967 original. · Zbl 1179.11040
[3] Laurent Berger, Représentations modulaires de \?\?\(_{2}\)(\?_\?) et représentations galoisiennes de dimension 2, Astérisque 330 (2010), 263 – 279 (French, with English and French summaries). · Zbl 1233.11060
[4] Don Blasius, Hilbert modular forms and the Ramanujan conjecture, Noncommutative geometry and number theory, Aspects Math., E37, Friedr. Vieweg, Wiesbaden, 2006, pp. 35 – 56. · Zbl 1183.11023 · doi:10.1007/978-3-8348-0352-8_2 · doi.org
[5] Tom Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor, Potential automorphy and change of weight, preprint available at http://www. math.northwestern.edu/ gee/, 2010. · Zbl 1310.11060
[6] Tom Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor, A family of Calabi-Yau varieties and potential automorphy II, Preprint, 2009. · Zbl 1264.11044
[7] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. · Zbl 0980.22015
[8] Brian Conrad, Fred Diamond, and Richard Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), no. 2, 521 – 567. · Zbl 0923.11085
[9] Laurent Clozel, Michael Harris, and Richard Taylor, Automorphy for some \?-adic lifts of automorphic mod \? Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1 – 181. With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras. · Zbl 1169.11020 · doi:10.1007/s10240-008-0016-1 · doi.org
[10] Laurent Clozel, Changement de base pour les représentations tempérées des groupes réductifs réels, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 1, 45 – 115 (French). · Zbl 0516.22010
[11] Mladen Dimitrov, Galois representations modulo \? and cohomology of Hilbert modular varieties, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 4, 505 – 551 (English, with English and French summaries). · Zbl 1160.11325 · doi:10.1016/j.ansens.2005.03.005 · doi.org
[12] P. Deligne, D. Kazhdan, and M.-F. Vignéras, Représentations des algèbres centrales simples \?-adiques, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 33 – 117 (French). · Zbl 0583.22009
[13] Toby Gee, A modularity lifting theorem for weight two Hilbert modular forms, Math. Res. Lett. 13 (2006), no. 5-6, 805 – 811. · Zbl 1185.11030 · doi:10.4310/MRL.2006.v13.n5.a10 · doi.org
[14] Toby Gee, The Sato-Tate conjecture for modular forms of weight 3, Doc. Math. 14 (2009), 771 – 800. · Zbl 1246.11100
[15] -, Automorphic lifts of prescribed types, to appear Math. Annalen, available at http://www.math.northwestern.edu/ gee/, 2010.
[16] David Geraghty, Modularity lifting theorems for ordinary Galois representations, preprint, 2009.
[17] Toby Gee and David Geraghty, Companion forms for unitary and symplectic groups, Preprint, 2009. · Zbl 1295.11043
[18] L. Guerberoff, Modularity lifting theorems for Galois representations of unitary type, Arxiv preprint arXiv:0906.4189 (2009).
[19] Michael Harris, Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progr. Math., vol. 270, Birkhäuser Boston, Inc., Boston, MA, 2009, pp. 1 – 21. · Zbl 1234.11068 · doi:10.1007/978-0-8176-4747-6_1 · doi.org
[20] Michael Harris, Nick Shepherd-Barron, and Richard Taylor, A family of Calabi-Yau varieties and potential automorphy, Ann. of Math. (2) 171 (2010), no. 2, 779 – 813. · Zbl 1263.11061 · doi:10.4007/annals.2010.171.779 · doi.org
[21] Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. · Zbl 1036.11027
[22] Mark Kisin, Modularity of 2-dimensional Galois representations, Current developments in mathematics, 2005, Int. Press, Somerville, MA, 2007, pp. 191 – 230. · Zbl 1218.11056
[23] Mark Kisin, Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2008), no. 2, 513 – 546. · Zbl 1205.11060
[24] -, Moduli of finite flat group schemes, and modularity, Annals of Math. (2) 170 (2009), no. 3, 1085-1180. · Zbl 1201.14034
[25] Jean-Pierre Labesse, Changement de base CM et séries discrètes, preprint, 2009.
[26] R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101 – 170. · Zbl 0741.22009 · doi:10.1090/surv/031/03 · doi.org
[27] David Savitt, On a conjecture of Conrad, Diamond, and Taylor, Duke Math. J. 128 (2005), no. 1, 141 – 197. · Zbl 1101.11017 · doi:10.1215/S0012-7094-04-12816-7 · doi.org
[28] Jean-Pierre Serre, Abelian \?-adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc., New York-Amsterdam, 1968. · Zbl 0186.25701
[29] Jean-Pierre Serre, Sur la semi-simplicité des produits tensoriels de représentations de groupes, Invent. Math. 116 (1994), no. 1-3, 513 – 530 (French). · Zbl 0816.20014 · doi:10.1007/BF01231571 · doi.org
[30] Richard Taylor, On the meromorphic continuation of degree two \?-functions, Doc. Math. Extra Vol. (2006), 729 – 779. · Zbl 1138.11051
[31] Richard Taylor, Automorphy for some \?-adic lifts of automorphic mod \? Galois representations. II, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 183 – 239. · Zbl 1169.11021 · doi:10.1007/s10240-008-0015-2 · doi.org
[32] Richard Taylor and Teruyoshi Yoshida, Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc. 20 (2007), no. 2, 467 – 493. · Zbl 1210.11118
[33] A. V. Zelevinsky, Induced representations of reductive \?-adic groups. II. On irreducible representations of \?\?(\?), Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 165 – 210. · Zbl 0441.22014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.