On wild ramification in quaternion extensions
Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 101-124.

Cet article fournit un catalogue complet des nombres de ramification qui se produisent dans la filtration de ramification des extensions totalement ramifiées des corps de nombres dyadiques qui contiennent -1, et dont le groupe Galois est isomorphe au groupe des quaternions (avec quelques résultats partiels dans le cas plus général). Ce catalogue dépend d’un rafinement de la filtration de ramification. Cette filtration était definie dans [2] comme associée au sous-corps biquadratique. En outre, nous montrons que les contre-exemples de type quaternion aux conclusions du théorème de Hasse-Arf sont extrêmement rares et ne peuvent se produire que seulement dans le cas où la filtration raffinée de ramification est extrême dans deux directions distinctes.

This paper provides a complete catalog of the break numbers that occur in the ramification filtration of fully and thus wildly ramified quaternion extensions of dyadic number fields which contain -1 (along with some partial results for the more general case). This catalog depends upon the refined ramification filtration, which as defined in [2] is associated with the biquadratic subfield. Moreover we find that quaternion counter-examples to the conclusion of the Hasse-Arf Theorem are extremely rare and can occur only when the refined ramification filtration is, in two different ways, extreme.

DOI : 10.5802/jtnb.576
Elder, G. Griffith 1 ; Hooper, Jeffrey J. 2

1 Department of Mathematics Virginia Tech Blacksburg, VA 24061-0123 U.S.A.
2 Department of Mathematics and Statistics Acadia University Wolfville, NS B4P 2R6 Canada
@article{JTNB_2007__19_1_101_0,
     author = {Elder, G. Griffith and Hooper, Jeffrey J.},
     title = {On wild ramification in quaternion extensions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {101--124},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     doi = {10.5802/jtnb.576},
     zbl = {1123.11037},
     mrnumber = {2332056},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.576/}
}
TY  - JOUR
AU  - Elder, G. Griffith
AU  - Hooper, Jeffrey J.
TI  - On wild ramification in quaternion extensions
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2007
SP  - 101
EP  - 124
VL  - 19
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.576/
DO  - 10.5802/jtnb.576
LA  - en
ID  - JTNB_2007__19_1_101_0
ER  - 
%0 Journal Article
%A Elder, G. Griffith
%A Hooper, Jeffrey J.
%T On wild ramification in quaternion extensions
%J Journal de théorie des nombres de Bordeaux
%D 2007
%P 101-124
%V 19
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.576/
%R 10.5802/jtnb.576
%G en
%F JTNB_2007__19_1_101_0
Elder, G. Griffith; Hooper, Jeffrey J. On wild ramification in quaternion extensions. Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 101-124. doi : 10.5802/jtnb.576. http://www.numdam.org/articles/10.5802/jtnb.576/

[1] N. P. Byott, G. G. Elder, Biquadratic extensions with one break. Canadian Math. Bull. 45 (2002) no. 2, 168–179. | MR | Zbl

[2] N. P. Byott, G. G. Elder, New ramification breaks and additive Galois structure. J. Théor. Nombres Bordeaux 17 (2005) no. 1, 87–07, Les XXIIIémes Journées Arithmetiques (Graz, 2003). | Numdam | MR | Zbl

[3] G. G. Elder, Galois module structure in wildly ramified cyclic extensions of degree p 2 . Ann. Inst. Fourier (Grenoble) 45 (1995) no. 3, 625–647. errata ibid. 48 (1998) no. 2, 609–610. | Numdam | Zbl

[4] G. G. Elder, Galois module structure of ambiguous ideals in biquadratic extensions. Canad. J. Math. 50 (1998), no. 5, 1007–1047. | MR | Zbl

[5] G. G. Elder, The Galois structure of ambiguous ideals in cyclic extensions of degree 8. Noncommutative algebra and geometry, 63–89, Lect. Notes Pure Appl. Math., 243, Chapman & Hall/CRC, Boca Raton, FL, 2006. | MR | Zbl

[6] I. B. Fesenko, S. V. Vostokov, Local fields and their extensions. Trans. of Math. Monographs, 121, 2nd Ed., American Mathematical Society, Providence, RI, 2002. | MR | Zbl

[7] J.-M.  Fontaine, Groupes de ramification et représentations d’Artin. Ann. Sci. École Norm. Sup. (4) 4 (1971), 337–392. | Numdam | Zbl

[8] A. Fröhlich, Galois module structure of algebraic integers. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 1, Springer-Verlag, Berlin, 1983. | MR | Zbl

[9] H. Hasse, Number theory. Classics in Mathematics, Springer-Verlag, Berlin, 2002, Reprint of 1980 English ed. Edited with a preface by H. G. Zimmer. | MR | Zbl

[10] C. U. Jensen, N. Yui, Quaternion extensions. Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 155–182. | MR | Zbl

[11] O. T. O’Meara, Introduction to quadratic forms. Springer-Verlag, New York, 1971. | Zbl

[12] H. Reichardt, Über normalkörper mit quaternionengruppe. Math. Z. 41 (1936), 218–221. | MR

[13] J.-P. Serre, Local Fields. Springer-Verlag, New York, 1979. | MR | Zbl

[14] E. Witt, Konstruktion von galoisschen körpern der characteristik p zu vorgegebener gruppe der ordnung p f . J. Reine Angew. Math. 174 (1936), 237–245.

[15] B. Wyman, Wildly ramified gamma extensions. Amer. J. Math. 91 (1969), 135–152. | MR | Zbl

Cité par Sources :