On modular forms and the inverse Galois problem
Authors:
Luis Dieulefait and Gabor Wiese
Journal:
Trans. Amer. Math. Soc. 363 (2011), 4569-4584
MSC (2010):
Primary 11F80; Secondary 12F12, 11F11
DOI:
https://doi.org/10.1090/S0002-9947-2011-05477-2
Published electronically:
April 11, 2011
MathSciNet review:
2806684
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this article new cases of the inverse Galois problem are established. The main result is that for a fixed integer , there is a positive density set of primes
such that
occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Particular care is taken in order to exclude nontrivial inner twists.
- [B] Armand Brumer, The rank of 𝐽₀(𝑁), Astérisque 228 (1995), 3, 41–68. Columbia University Number Theory Seminar (New York, 1992). MR 1330927
- [DT] Fred Diamond and Richard Taylor, Nonoptimal levels of mod 𝑙 modular representations, Invent. Math. 115 (1994), no. 3, 435–462. MR 1262939, https://doi.org/10.1007/BF01231768
- [Di1] Luis V. Dieulefait, Newforms, inner twists, and the inverse Galois problem for projective linear groups, J. Théor. Nombres Bordeaux 13 (2001), no. 2, 395–411 (English, with English and French summaries). MR 1879665
- [Di2] Luis Dieulefait, A control theorem for the images of Galois actions on certain infinite families of modular forms, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, 2008, pp. 79–83. MR 2530979, https://doi.org/10.1017/CBO9780511543371.007
- [Di3] L. V. Dieulefait. Remarks on Serre's modularity conjecture. Preprint (2006), arXiv: math/ 0603439.
- [DV] Luis Dieulefait and Núria Vila, Projective linear groups as Galois groups over 𝑄 via modular representations, J. Symbolic Comput. 30 (2000), no. 6, 799–810. Algorithmic methods in Galois theory. MR 1800679, https://doi.org/10.1006/jsco.1999.0383
- [KLS] Chandrashekhar Khare, Michael Larsen, and Gordan Savin, Functoriality and the inverse Galois problem, Compos. Math. 144 (2008), no. 3, 541–564. MR 2422339, https://doi.org/10.1112/S0010437X07003284
- [KLS2] Chandrashekhar Khare, Michael Larsen, and Gordan Savin, Functoriality and the inverse Galois problem. II. Groups of type 𝐵_{𝑛} and 𝐺₂, Ann. Fac. Sci. Toulouse Math. (6) 19 (2010), no. 1, 37–70 (English, with English and French summaries). MR 2597780
- [KW0] Chandrashekhar Khare and Jean-Pierre Wintenberger, On Serre’s conjecture for 2-dimensional mod 𝑝 representations of 𝐺𝑎𝑙(\overline{ℚ}/ℚ), Ann. of Math. (2) 169 (2009), no. 1, 229–253. MR 2480604, https://doi.org/10.4007/annals.2009.169.229
- [KW1] Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504. MR 2551763, https://doi.org/10.1007/s00222-009-0205-7
- [KW2] Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586. MR 2551764, https://doi.org/10.1007/s00222-009-0206-6
- [Ki] Mark Kisin, Modularity of 2-adic Barsotti-Tate representations, Invent. Math. 178 (2009), no. 3, 587–634. MR 2551765, https://doi.org/10.1007/s00222-009-0207-5
- [M] Daniel A. Marcus, Number fields, Springer-Verlag, New York-Heidelberg, 1977. Universitext. MR 0457396
- [Q] Jordi Quer, Liftings of projective 2-dimensional Galois representations and embedding problems, J. Algebra 171 (1995), no. 2, 541–566. MR 1315912, https://doi.org/10.1006/jabr.1995.1027
- [RV] Amadeu Reverter and Núria Vila, Some projective linear groups over finite fields as Galois groups over 𝑄, Recent developments in the inverse Galois problem (Seattle, WA, 1993) Contemp. Math., vol. 186, Amer. Math. Soc., Providence, RI, 1995, pp. 51–63. MR 1352266, https://doi.org/10.1090/conm/186/02175
- [R1] Kenneth A. Ribet, On 𝑙-adic representations attached to modular forms, Invent. Math. 28 (1975), 245–275. MR 419358, https://doi.org/10.1007/BF01425561
- [R2] Kenneth A. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980), no. 1, 43–62. MR 594532, https://doi.org/10.1007/BF01457819
- [R3] Kenneth A. Ribet, On 𝑙-adic representations attached to modular forms. II, Glasgow Math. J. 27 (1985), 185–194. MR 819838, https://doi.org/10.1017/S0017089500006170
- [R4] Kenneth A. Ribet, Images of semistable Galois representations, Pacific J. Math. Special Issue (1997), 277–297. Olga Taussky-Todd: in memoriam. MR 1610883, https://doi.org/10.2140/pjm.1997.181.277
- [T] J. Tate, Number theoretic background, Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607
- [We] Jared Weinstein, Hilbert modular forms with prescribed ramification, Int. Math. Res. Not. IMRN 8 (2009), 1388–1420. MR 2496768, https://doi.org/10.1093/imrn/rnn161
- [Wi] Gabor Wiese, On projective linear groups over finite fields as Galois groups over the rational numbers, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, 2008, pp. 343–350. MR 2530980, https://doi.org/10.1017/CBO9780511543371.018
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11F80, 12F12, 11F11
Retrieve articles in all journals with MSC (2010): 11F80, 12F12, 11F11
Additional Information
Luis Dieulefait
Affiliation:
Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
Email:
ldieulefait@ub.edu
Gabor Wiese
Affiliation:
Institut für Experimentelle Mathematik, Universität Duisburg-Essen, Ellernstraße 29, 45326 Essen, Germany
Email:
gabor.wiese@uni-due.de
DOI:
https://doi.org/10.1090/S0002-9947-2011-05477-2
Keywords:
Modular forms,
Galois representations,
inverse Galois problem
Received by editor(s):
May 26, 2009
Published electronically:
April 11, 2011
Additional Notes:
The first author was partially supported by the grant MTM2009-07024 from the Ministerio de Ciencia e Innovación (Spain).
The second author acknowledges partial support by the Sonderforschungsbereich Transregio 45 of the Deutsche Forschungsgemeinschaft. Both authors were partially supported by the European Research Training Network Galois Theory and Explicit Methods MRTN-CT-2006-035495.
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.