Geodesic
The Sato-Tate conjecture for Hilbert modular forms
Barnet-Lamb, Thomas1 ; Gee, Toby2, 3 ; Geraghty, David2, 4
1 Department of Mathematics, Brandeis University, 415 South Street MS 050, Waltham, Massachusetts 02138
2 Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
3 Department of Mathematics, Northwestern University, 2033 Sheridan Road Evanston, Ilinois 60208-2730
4 Princeton University and Institute for Advanced Study, Princeton, New Jersey 08540
Journal of the American Mathematical Society, Tome 24 (2011) no. 2, pp. 411-469

Voir la notice de l'article provenant de la source American Mathematical Society

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of $\operatorname {GL}_2(\mathbb {A}_F)$, $F$ a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et al., but makes use of automorphy lifting theorems over ramified fields, together with a “topological” argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of potential automorphy theorems for non-ordinary $n$-dimensional Galois representations.
DOI : 10.1090/S0894-0347-2010-00689-3

Barnet-Lamb, Thomas 1 ; Gee, Toby 2, 3 ; Geraghty, David 2, 4

1 Department of Mathematics, Brandeis University, 415 South Street MS 050, Waltham, Massachusetts 02138
2 Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
3 Department of Mathematics, Northwestern University, 2033 Sheridan Road Evanston, Ilinois 60208-2730
4 Princeton University and Institute for Advanced Study, Princeton, New Jersey 08540 
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Barnet-Lamb, Thomas; Gee, Toby; Geraghty, David. The Sato-Tate conjecture for Hilbert modular forms. Journal of the American Mathematical Society, Tome 24 (2011) no. 2, pp. 411-469. doi: 10.1090/S0894-0347-2010-00689-3

[1] Arthur, James, Clozel, Laurent Simple algebras, base change, and the advanced theory of the trace formula 1989

[2] Artin, Emil, Tate, John Class field theory 2009

[3] Berger, Laurent Représentations modulaires de 𝐺𝐿₂(𝐐_{𝐩}) et représentations galoisiennes de dimension 2 Astérisque 2010 263 279

[4] Blasius, Don Hilbert modular forms and the Ramanujan conjecture 2006 35 56

[5] Borel, A., Wallach, N. Continuous cohomology, discrete subgroups, and representations of reductive groups 2000

[6] Conrad, Brian, Diamond, Fred, Taylor, Richard Modularity of certain potentially Barsotti-Tate Galois representations J. Amer. Math. Soc. 1999 521 567

[7] Clozel, Laurent, Harris, Michael, Taylor, Richard Automorphy for some 𝑙-adic lifts of automorphic mod 𝑙 Galois representations Publ. Math. Inst. Hautes Études Sci. 2008 1 181

[8] Clozel, Laurent Changement de base pour les représentations tempérées des groupes réductifs réels Ann. Sci. École Norm. Sup. (4) 1982 45 115

[9] Dimitrov, Mladen Galois representations modulo 𝑝 and cohomology of Hilbert modular varieties Ann. Sci. École Norm. Sup. (4) 2005 505 551

[10] Deligne, P., Kazhdan, D., Vignã©Ras, M.-F. Représentations des algèbres centrales simples 𝑝-adiques 1984 33 117

[11] Gee, Toby A modularity lifting theorem for weight two Hilbert modular forms Math. Res. Lett. 2006 805 811

[12] Gee, Toby The Sato-Tate conjecture for modular forms of weight 3 Doc. Math. 2009 771 800

[13] Harris, Michael Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications 2009 1 21

[14] Harris, Michael, Shepherd-Barron, Nick, Taylor, Richard A family of Calabi-Yau varieties and potential automorphy Ann. of Math. (2) 2010 779 813

[15] Harris, Michael, Taylor, Richard The geometry and cohomology of some simple Shimura varieties 2001

[16] Kisin, Mark Modularity of 2-dimensional Galois representations 2007 191 230

[17] Kisin, Mark Potentially semi-stable deformation rings J. Amer. Math. Soc. 2008 513 546

[18] Langlands, R. P. On the classification of irreducible representations of real algebraic groups 1989 101 170

[19] Savitt, David On a conjecture of Conrad, Diamond, and Taylor Duke Math. J. 2005 141 197

[20] Serre, Jean-Pierre Abelian 𝑙-adic representations and elliptic curves 1968

[21] Serre, Jean-Pierre Sur la semi-simplicité des produits tensoriels de représentations de groupes Invent. Math. 1994 513 530

[22] Taylor, Richard On the meromorphic continuation of degree two 𝐿-functions Doc. Math. 2006 729 779

[23] Taylor, Richard Automorphy for some 𝑙-adic lifts of automorphic mod 𝑙 Galois representations. II Publ. Math. Inst. Hautes Études Sci. 2008 183 239

[24] Taylor, Richard, Yoshida, Teruyoshi Compatibility of local and global Langlands correspondences J. Amer. Math. Soc. 2007 467 493

[25] Zelevinsky, A. V. Induced representations of reductive 𝔭-adic groups. II. On irreducible representations of 𝔊𝔏(𝔫) Ann. Sci. École Norm. Sup. (4) 1980 165 210

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