Companion forms for unitary and symplectic groups. (English) Zbl 1295.11043

From the text: “The problem of companion forms was first introduced by J.-P. Serre [ibid. J. 54, 179–230 (1987; Zbl 0641.10026)] for modular forms in his seminal paper. Fix a prime \(l\), algebraic closures \(\overline{\mathbb Q}\) and \(\overline{\mathbb Q_l}\) of \(\mathbb Q\) and \(\mathbb Q_l\), respectively, and an embedding of \(\overline{\mathbb Q}\) into \(\overline{\mathbb Q_l}\). Suppose that \(f\) is a modular newform of weight \(k> 2\) which is ordinary at \(l\), so that the corresponding \(l\)-adic Galois representation \(\rho_{f,l}\) becomes reducible when restricted to a decomposition group \(G_{\mathbb Q_l}\) at \(l\). Then the companion forms problem is essentially the question of determining for which other weights \(k'\) there is an ordinary newform \(g\) of weight \(k'\geq 2\) such that the Galois representations \(\rho_{f,l}\) and \(\rho_{g,l}\) are congruent modulo \(l\). The problem is straightforward unless the restriction to \(G_{\mathbb Q_l}\) of \(\overline\rho_{f,l}\) (the reduction mod \(l\) of \(\rho_{f,l}\)) is split and nonscalar, in which case there are two possible Hida families whose corresponding Galois representations lift \(\overline\rho_{f,l}\). The restrictions of the corresponding Galois representations to a decomposition group at \(l\) are either “upper triangular” or “lower triangular”.”
“This problem was essentially resolved by B. H. Gross [ibid. 61, No. 2, 445–517 (1990; Zbl 0743.11030)] and R. F. Coleman and J. F. Voloch [Invent. Math. 110, No. 2, 263–281 (1992; Zbl 0770.11024)]. In the paper [ibid. 136, No. 2, 275–320 (2007; Zbl 1121.11039)], the first author reproved these results and generalized them to Hilbert modular forms, by a completely new technique.”
“We prove a companion forms theorem for ordinary \(n\)-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author and a technique for deducing companion forms theorems developed by the first author. We deduce results about the possible Serre weights of \(\mod l\) Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of \(\mathrm{GSp}_4\) over totally real fields.”


11F33 Congruences for modular and \(p\)-adic modular forms
11F55 Other groups and their modular and automorphic forms (several variables)
11F80 Galois representations
Full Text: DOI arXiv


[1] J. Arthur, “Automorphic representations of GSp(4)” in Contributions to Automorphic Forms, Geometry, and Number Theory , Johns Hopkins Univ. Press, Baltimore, 2004, 65-81. · Zbl 1080.11037
[2] A. Ash, D. Doud, and D. Pollack, Galois representations with conjectural connections to arithmetic cohomology , Duke Math. J. 112 (2002), 521-579. · Zbl 1023.11025
[3] T. Barnet-Lamb, T. Gee, D. Geraghty, and R. Taylor, Potential automorphy and change of weight , preprint,
[4] T. Barnet-Lamb, D. Geraghty, M. Harris, and R. Taylor, A family of Calabi-Yau varieties and potential automorphy , II , Publ. Res. Inst. Math. Sci. 47 (2011), 29-98. · Zbl 1264.11044
[5] H. Carayol, “Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet” in p - adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, 1991) , Contemp. Math. 165 , Amer. Math. Soc., Providence, 1994, 213-237. · Zbl 0812.11036
[6] G. Chenevier and M. Harris, Construction of automorphic Galois representations, II , preprint, 2009. · Zbl 1310.11062
[7] L. Clozel, M. Harris, and R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations , Publ. Math. Inst. Haute Études Sci. 108 (2008), 1-181. · Zbl 1169.11020
[8] R. F. Coleman and J. F. Voloch, Companion forms and Kodaira-Spencer theory , Invent. Math. 110 (1992), 263-281. · Zbl 0770.11024
[9] A. J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry , Inst. Hautes Études Sci. Publ. Math. 82 (1995), 5-96. · Zbl 0864.14009
[10] T. Gee, Companion forms over totally real fields, II , Duke Math. J. 136 (2007), 275-284. · Zbl 1121.11039
[11] T. Gee, Automorphic lifts of prescribed types , Math. Ann. 350 (2011), 107-144. · Zbl 1276.11085
[12] D. Geraghty, Modularity lifting theorems for ordinary Galois representations , preprint, 2009, available at
[13] B. H. Gross, A tameness criterion for Galois representations associated to modular forms (mod p ), Duke Math. J. 61 (1990), 445-517. · Zbl 0743.11030
[14] A. Genestier and J. Tilouine, “Systèmes de Taylor-Wiles pour GSp 4 ” in Formes automorphes, II: Le cas du groupe GSp(4), Astérisque 302 , Soc. Math. France, Montrouge, 2005, 177-290. · Zbl 1142.11036
[15] W. T. Gan and S. Takeda, The local Langlands conjecture for GSp(4), Ann. of Math. (2) 173 (2011), 1841-1882. · Zbl 1230.11063
[16] F. Herzig, The weight in a Serre-type conjecture for tame n-dimensional Galois representations , Duke Math. J. 149 (2009), 37-116. · Zbl 1232.11065
[17] M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties , Ann. Math. Stud. 151 , Princeton Univ. Press, Princeton, 2001. · Zbl 1036.11027
[18] F. Herzig and J. Tilouine, Conjecture de type Serre et formes compagnons pour GSp(4), to appear in J. Reine Agnew. Math., preprint,
[19] J. C. Jantzen, Representations of Algebraic Groups , 2nd ed., Math. Surv. Monogr. 107 , Amer. Math. Soc., Providence, 2003. · Zbl 1034.20041
[20] M. Kisin, “Modularity of 2-dimensional Galois representations” in Current Developments in Mathematics , 2005, Int. Press, Somerville, Mass., 2007, 191-230. · Zbl 1218.11056
[21] M. Kisin, Potentially semi-stable deformation rings , J. Amer. Math. Soc. 21 (2008), 513-546. · Zbl 1205.11060
[22] M. Kisin, Moduli of finite flat group schemes, and modularity , Ann. of Math. (2) 170 (2009), 1085-1180. · Zbl 1201.14034
[23] C. Khare and J.-P. Wintenberger, On Serre’s conjecture for 2- dimensional mod p representations of the absolute Galois group of the rationals , Ann. of Math. (2) 169 (2009), 229-253.
[24] J.-P. Labesse, Changement de base CM et séries discrètes , preprint, 2009.
[25] B. Mazur, “Deforming Galois representations” in Galois Groups Over Q (Berkeley, Calif., 1987) , Math. Sci. Res. Inst. Publ. 16 , Springer, New York, 1989, 385-437. · Zbl 0714.11076
[26] J. Nekovář, “On p -adic height pairings” in Séminaire de Théorie des Nombres (Paris, 1990-91) , Progr. Math. 108 , Birkhäuser, Boston, 1993, 127-202.
[27] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal ( Q\? / Q ), Duke Math. J. 54 (1987), 179-230. · Zbl 0641.10026
[28] C. Sorensen, Galois representations and Hilbert-Siegel modular forms , Doc. Math. 15 (2010), 623-670. · Zbl 1246.11114
[29] R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations, II , Publ. Math. Inst. Haute Etudes Études Sci. 108 (2008), 183-239. · Zbl 1169.11021
[30] J. Thorne, On the automorphy of l-adic Galois representations with small residual image , preprint, 2010, available at
[31] J. Tilouine, Deformations of Galois Representations and Hecke Algebras , published for The Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, Narosa Publishing House, New Delhi, 1996. · Zbl 1009.11033
[32] J. Tilouine, Formes compagnons et complexe BGG dual pour GSp(4), preprint, 2009.
[33] R. Weissauer, “Four dimensional Galois representations” in Formes automorphes, II : Le cas du groupe GSp(4), Astérisque 302 , Soc. Math. France, Montrouge, 2005, 67-150. · Zbl 1097.11027
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.