Compositio Mathematica

Research Article

On number fields with given ramification

Gaëtan Cheneviera1

a1 Institut Galilée LAGA, Université Paris 13, 99 Avenue J-B. Clément, 93430 Villetaneuse, France (email:,


Let E be a CM number field and let S be a finite set of primes of E containing the primes dividing a given prime number l and another prime u split above the maximal totally real subfield of E. If ES denotes a maximal algebraic extension of E which is unramified outside S, we show that the natural maps $\mathrm {Gal}(\overline {E_u}/E_u) \longrightarrow \mathrm {Gal}(E_S/E)$ are injective. We discuss generalizations of this result.

(Received March 12 2007)

(Accepted May 21 2007)

(Online publication November 09 2007)


  • global Galois groups;
  • restricted ramification;
  • automorphic forms

2000 Mathematics subject classification

  • 11F55;
  • 11F80;
  • 11R32


The author is supported by the C.N.R.S.