Comptes Rendus
Number Theory
A non-solvable Galois extension of Q ramified at 2 only
Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 111-116.

In this Note, we show the existence of a non-solvable Galois extension of Q which is unramified outside 2. The extension K we construct has degree 2251731094732800=219(3517257)2, it has root discriminant δK<2478=58.68 , and is totally complex.

Dans cette Note, nous démontrons l'existence d'une extension galoisienne non résoluble de Q ramifiée seulement en 2. L'extension K que nous construisons est de degré 2251731094732800=219(3517257)2 et de discriminant normalisé δK<2478=58,68 , et est totalement complexe.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.12.004

Lassina Dembélé 1

1 Institut für Experimentelle Mathematik, Ellernstrasse 29, 45141 Essen, Germany
@article{CRMATH_2009__347_3-4_111_0,
     author = {Lassina Demb\'el\'e},
     title = {A non-solvable {Galois} extension of $ \mathbb{Q}$ ramified at 2 only},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {111--116},
     publisher = {Elsevier},
     volume = {347},
     number = {3-4},
     year = {2009},
     doi = {10.1016/j.crma.2008.12.004},
     language = {en},
}
TY  - JOUR
AU  - Lassina Dembélé
TI  - A non-solvable Galois extension of $ \mathbb{Q}$ ramified at 2 only
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 111
EP  - 116
VL  - 347
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crma.2008.12.004
LA  - en
ID  - CRMATH_2009__347_3-4_111_0
ER  - 
%0 Journal Article
%A Lassina Dembélé
%T A non-solvable Galois extension of $ \mathbb{Q}$ ramified at 2 only
%J Comptes Rendus. Mathématique
%D 2009
%P 111-116
%V 347
%N 3-4
%I Elsevier
%R 10.1016/j.crma.2008.12.004
%G en
%F CRMATH_2009__347_3-4_111_0
Lassina Dembélé. A non-solvable Galois extension of $ \mathbb{Q}$ ramified at 2 only. Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 111-116. doi : 10.1016/j.crma.2008.12.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.12.004/

[1] W. Bosma; J. Cannon; C. Playoust The Magma algebra system. I. The user language, J. Symbolic Comput., Volume 24 (1997) no. 3–4, pp. 235-265

[2] S. Brueggeman The nonexistence of certain Galois extensions unramified outside 5, J. Number Theory, Volume 75 (1999), pp. 47-52

[3] C. Breuil; B. Conrad; F. Diamond; R. Taylor On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc., Volume 14 (2001) no. 4, pp. 843-939

[4] H. Carayol Sur les représentations l-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Ecole Norm. Sup., Volume 19 (1986), pp. 409-468

[5] F. Diaz y Diaz, Tables minorant la racine n-ième du discriminant d'un corps de degré n, Publications Mathématiques d'Orsay 80, Université de Paris-Sud, Département de Mathématique, Orsay, 1980, 59 pp

[6] B. Gross Modular forms (mod p) and Galois representations, Int. Math. Res. Notices, Volume 16 (1998), pp. 865-875

[7] B. Gross Algebraic modular forms, Israel J. Math., Volume 113 (1999), pp. 61-93

[8] F. Hajir; C. Maire Tamely ramified towers and discriminant bounds for number fields. II, J. Symbolic Comput., Volume 33 (2002) no. 4, pp. 415-423

[9] F. Hajir; C. Maire Tamely ramified towers and discriminant bounds for number fields, Compositio Math., Volume 128 (2001) no. 1, pp. 35-53

[10] F. Jarvis On Galois representations associated to Hilbert modular forms of low weight, J. Reine Angew. Math., Volume 491 (1997), pp. 199-216

[11] C. Khare, J.-P. Wintenberger, On Serre's conjecture for 2-dimensional mod p representations of the absolute Galois group of the rationals, Ann. of Math., in press

[12] J. Lansky; D. Pollack Hecke algebras and automorphic forms, Compositio Math., Volume 130 (2002) no. 1, pp. 21-48

[13] M. Ohta Hilbert modular forms of weight one and Galois representations, Progr. Math., Volume 46 (1984), pp. 333-353

[14] J. Rogawski; J. Tunnell On Artin L-functions associated to Hilbert modular forms of weight 1, Invent. Math., Volume 74 (1983), pp. 1-42

[15] J.-P. Serre Sur les représentations modulaires de degré 2 de Gal(Q¯/Q), Duke Math. J., Volume 54 (1987) no. 1, pp. 179-230

[16] J.-P. Serre Corps locaux, Publications de l'Université de Nancago, vol. VIII, Hermann, Paris, 1968, p. 245

[17] J.-P. Serre Congruences et formes modulaires [d'après H.P.F. Swinnerton-Dyer], Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 416, Lecture Notes in Math., vol. 317, Springer, Berlin, 1973, pp. 319-338

[18] J.-P. Serre Oeuvres III, Springer-Verlag, 1986 (Note 229.2 on p. 710)

[19] J.-P. Serre Abelian l-Adic Representations and Elliptic Curves, Research Notes in Mathematics, vol. 7, A K Peters, Ltd., Wellesley, MA, 1997

[20] J. Tate The non-existence of certain Galois extensions of Q unramified outside 2, Contemp. Math., Volume 174 (1994), pp. 153-156

[21] R. Taylor On Galois representations associated to Hilbert modular forms, Invent. Math., Volume 98 (1989) no. 2, pp. 265-280

[22] R. Taylor On the meromorphic continuation of degree two L-functions, Doc. Math. Extra Vol. (2006), pp. 729-779

[23] A. Wiles On ordinary λ-adic representations associated to modular forms, Invent. Math., Volume 94 (1988) no. 3, pp. 529-573

Cited by Sources:

Comments - Policy