# 5 avril 2018: 1 événement

### Jeudi 5 avril 14:00-15:30 - Jishnu Ray - Laboratoire de Mathématiques d'Orsay

Sém. ATDN - Iwasawa algebras of $p$-adic Lie groups and Galois representations with open image

Résumé : A key tool in the study of algebraic number fields are Iwasawa algebras, originally constructed by Iwasawa in the 1960’s to study the "class groups" of fields, but since appearing in varied settings such as a Lazard’s work on p-adic Lie groups and Fontaine’s work on local Galois representations. For a prime p, the Iwasawa algebra of a p-adic Lie group G, denoted by Zp[ [G] ], is a non-commutative completed group algebra of G.
In the first part of the talk, we lay the foundation by giving a very explicit description of certain Iwasawa algebras (one such algebra was described by my advisor Clozel). The base change map between the Iwasawa algebras over extensions of Qp motivates us to discuss globally analytic p-adic representations following Emerton’s work.
In the second part of the talk, we will discuss about numerical experiments using a computer algebra system which give heuristic support to Greenberg’s p-rationality conjecture which affirms the existence of “p-rational” number fields with Galois groups (Z/2Z)^t\$. The p-rational fields are algebraic number fields whose Galois cohomology is particularly simple and which are interesting because they offer ways of constructing Galois representations with big open images. We go beyond Greenberg’s work and construct novel Galois representations of the absolute Galois group of Q with big open images in reductive groups over Zp (ex. GL(n ;Zp) ; SL(n ;Zp) ; SO(n ;Zp) ; Sp(n ;Zp)). We are proving results which show the existence of p-adic Lie extensions of Q where the Galois group corresponds to a certain specific p-adic Lie algebra (ex. sl(n) ; so(n) ; sp(2n)). This relates our work with a more general and classical Inverse Galois problem for p-adic Lie extensions.

Lieu : 324-2B

En savoir plus : Séminaire et groupe de travail d’Algèbre et Théorie des Nombres