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A note on free pro- p -extensions of algebraic number fields

Masakazu Yamagishi (1993)

Journal de théorie des nombres de Bordeaux

For an algebraic number field k and a prime p , define the number ρ to be the maximal number d such that there exists a Galois extension of k whose Galois group is a free pro- p -group of rank d . The Leopoldt conjecture implies 1 ρ r 2 + 1 , ( r 2 denotes the number of complex places of k ). Some examples of k and p with ρ = r 2 + 1 have been known so far. In this note, the invariant ρ is studied, and among other things some examples with ρ < r 2 + 1 are given.

Analogues étales de la p -tour des corps de classes

Jilali Assim (2003)

Journal de théorie des nombres de Bordeaux

Nous construisons un analogue «tordu» de la p -tour de corps de classes d’un corps de nombres ( p un nombre premier) et étudions ses liens avec la théorie d’Iwasawa. Le résultat principal donne un critère du type Golod et Shafarevich pour que la tour «tordue» soit infinie.

Automorphy for some l-adic lifts of automorphic mod l Galois representations. II

Richard Taylor (2008)

Publications Mathématiques de l'IHÉS

We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spirit to the methods of [TW] than to those of [W], which relied on Ihara’s lemma.

Automorphy for some l-adic lifts of automorphic mod l Galois representations

Laurent Clozel, Michael Harris, Richard Taylor (2008)

Publications Mathématiques de l'IHÉS

We extend the methods of Wiles and of Taylor and Wiles from GL2 to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), minimally ramified, l-adic lifts of certain automorphic mod l Galois representations of any dimension. We also make a conjecture about the structure of mod l automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for GL2. Following Wiles’ method we show that this...

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