Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


Class groups in cyclic $ \ell$-extensions: Comments on a paper by G. Cornell

Author: Michael Rosen
Journal: Proc. Amer. Math. Soc. 142 (2014), 21-28
MSC (2010): Primary 11R29
Published electronically: September 10, 2013
MathSciNet review: 3119177
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ E/F$ be a cyclic extension of number fields of degree $ \ell ^{n}$, where $ \ell $ is a prime. It is proved that the $ \ell $-rank of the class group of $ E$ is bounded by $ \ell ^{n}(t-1+ \mathrm {rk}_\ell Cl_{F})$, where $ t$ is the number of primes of $ F$, including infinite primes, which ramify in $ E$ and $ Cl_{F}$ is the class group of $ F$. This generalizes a result of G. Cornell which applies when $ n=1$ and $ \ell $ is odd. A similar result is shown to hold when the Galois group is an abelian $ \ell $-group and the Hasse norm theorem is valid for $ E/F$.

References [Enhancements On Off] (What's this?)

  • [1] R. Bröker, D. Gruenewald, and K. Lauter, Explicit CM theory for level 2-structures on abelian surfaces, Algebra Number Theory 5 (2011), 455-528. MR 2870099
  • [2] Gary Cornell, Relative genus theory and the class group of $ l$-extensions, Trans. Amer. Math. Soc. 277 (1983), no. 1, 421-429. MR 690061 (84i:12002),
  • [3] Yoshiomi Furuta, Über die Zentrale Klassenzahl eines relativ-galoisschen Zahlkörpers, J. Number Theory 3 (1971), 318-322 (German, with English summary). MR 0297743 (45 #6795)
  • [4] Dennis A. Garbanati, The Hasse norm theorem for non-cyclic extensions of the rationals, Proc. London Math. Soc. (3) 37 (1978), no. 1, 143-164. MR 0491599 (58 #10822)
  • [5] Michael J. Razar, Central and genus class fields and the Hasse norm theorem, Compositio Math. 35 (1977), no. 3, 281-298. MR 0466073 (57 #5956)
  • [6] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237 (82e:12016)
  • [7] J. T. Tate, Global class field theory, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 162-203. MR 0220697 (36 #3749)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11R29

Retrieve articles in all journals with MSC (2010): 11R29

Additional Information

Michael Rosen
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912

Keywords: $\ell$-rank, class groups, genus field, central class field
Received by editor(s): March 9, 2011
Received by editor(s) in revised form: December 21, 2011, and February 23, 2012
Published electronically: September 10, 2013
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society