## On modular forms and the inverse Galois problem

HTML articles powered by AMS MathViewer

- by Luis Dieulefait and Gabor Wiese
- Trans. Amer. Math. Soc.
**363**(2011), 4569-4584 - DOI: https://doi.org/10.1090/S0002-9947-2011-05477-2
- Published electronically: April 11, 2011
- PDF | Request permission

## Abstract:

In this article new cases of the inverse Galois problem are established. The main result is that for a fixed integer $n$, there is a positive density set of primes $p$ such that $\mathrm {PSL}_2(\mathbb {F}_{p^n})$ 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.## References

- Armand Brumer,
*The rank of $J_0(N)$*, Astérisque**228**(1995), 3, 41–68. Columbia University Number Theory Seminar (New York, 1992). MR**1330927** - Fred Diamond and Richard Taylor,
*Nonoptimal levels of mod $l$ modular representations*, Invent. Math.**115**(1994), no. 3, 435–462. MR**1262939**, DOI 10.1007/BF01231768 - 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** - 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**, DOI 10.1017/CBO9780511543371.007 - L. V. Dieulefait.
*Remarks on Serre’s modularity conjecture*. Preprint (2006), arXiv: math/ 0603439. - Luis Dieulefait and Núria Vila,
*Projective linear groups as Galois groups over $\textbf {Q}$ via modular representations*, J. Symbolic Comput.**30**(2000), no. 6, 799–810. Algorithmic methods in Galois theory. MR**1800679**, DOI 10.1006/jsco.1999.0383 - Chandrashekhar Khare, Michael Larsen, and Gordan Savin,
*Functoriality and the inverse Galois problem*, Compos. Math.**144**(2008), no. 3, 541–564. MR**2422339**, DOI 10.1112/S0010437X07003284 - Chandrashekhar Khare, Michael Larsen, and Gordan Savin,
*Functoriality and the inverse Galois problem. II. Groups of type $B_n$ and $G_2$*, Ann. Fac. Sci. Toulouse Math. (6)**19**(2010), no. 1, 37–70 (English, with English and French summaries). MR**2597780** - Chandrashekhar Khare and Jean-Pierre Wintenberger,
*On Serre’s conjecture for 2-dimensional mod $p$ representations of $\textrm {Gal}(\overline {\Bbb Q}/\Bbb Q)$*, Ann. of Math. (2)**169**(2009), no. 1, 229–253. MR**2480604**, DOI 10.4007/annals.2009.169.229 - Chandrashekhar Khare and Jean-Pierre Wintenberger,
*Serre’s modularity conjecture. I*, Invent. Math.**178**(2009), no. 3, 485–504. MR**2551763**, DOI 10.1007/s00222-009-0205-7 - Chandrashekhar Khare and Jean-Pierre Wintenberger,
*Serre’s modularity conjecture. II*, Invent. Math.**178**(2009), no. 3, 505–586. MR**2551764**, DOI 10.1007/s00222-009-0206-6 - Mark Kisin,
*Modularity of 2-adic Barsotti-Tate representations*, Invent. Math.**178**(2009), no. 3, 587–634. MR**2551765**, DOI 10.1007/s00222-009-0207-5 - Daniel A. Marcus,
*Number fields*, Universitext, Springer-Verlag, New York-Heidelberg, 1977. MR**0457396** - Jordi Quer,
*Liftings of projective $2$-dimensional Galois representations and embedding problems*, J. Algebra**171**(1995), no. 2, 541–566. MR**1315912**, DOI 10.1006/jabr.1995.1027 - Amadeu Reverter and Núria Vila,
*Some projective linear groups over finite fields as Galois groups over $\textbf {Q}$*, 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**, DOI 10.1090/conm/186/02175 - Kenneth A. Ribet,
*On $l$-adic representations attached to modular forms*, Invent. Math.**28**(1975), 245–275. MR**419358**, DOI 10.1007/BF01425561 - Kenneth A. Ribet,
*Twists of modular forms and endomorphisms of abelian varieties*, Math. Ann.**253**(1980), no. 1, 43–62. MR**594532**, DOI 10.1007/BF01457819 - Kenneth A. Ribet,
*On $l$-adic representations attached to modular forms. II*, Glasgow Math. J.**27**(1985), 185–194. MR**819838**, DOI 10.1017/S0017089500006170 - Kenneth A. Ribet,
*Images of semistable Galois representations*, Pacific J. Math.**Special Issue**(1997), 277–297. Olga Taussky-Todd: in memoriam. MR**1610883**, DOI 10.2140/pjm.1997.181.277 - 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** - Jared Weinstein,
*Hilbert modular forms with prescribed ramification*, Int. Math. Res. Not. IMRN**8**(2009), 1388–1420. MR**2496768**, DOI 10.1093/imrn/rnn161 - 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**, DOI 10.1017/CBO9780511543371.018

## Bibliographic 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
- MR Author ID: 671876
- 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
- 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. - © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2806684