Jean Pierre Wintenberger

Jean-Pierre Wintenberger

Description

Niveau: Supérieur, Doctorat, Bac+8
Jean-Pierre Wintenberger Extensions of Iwasawa modules (2) jw with Shekhar Khare The aim of the talk is show how Leopoldt conjecture is linked with prop- erties of exact sequences of Iwasawa modules arising from ramification at auxiliary primes. The hope is to be able to use modular technics to study these properties. We restrict to the case of a totally real F . Let p > 2 be a prime. Let F∞ = F (µp∞) be the cyclotomic extension. The cyclotomic character ?p identifies Gal(F∞/F ) to an open subgroup of (Zp)?, hence the quotient by its torsion is isomorphic to Zp. To this quotient, corresponds the cyclotomic Zp-extension we call F∞. A formulation of Leopoldt conjecture (LC) is that F∞ is the only Zp- extension of F . Let EF be the group of units of F . Let UF be the units in the p-adic completion of OF , so UF = ∏ ? U? where the ? are the primes of F over p. The group UF is the product the multiplicative groups ∏ ?(k?) ? of the residue fileds by the group U1F of units that reduces to 1 in ∏ ? k?. U 1 F is a Zp-module of rank r where r = [F : Q].

• iwasawa


• galois group over


• let m∞


• kummer extension


• residue fields


• adic modular


• galois over


• y∞ ?


• ?i reduce

Sujets

• Iwasawa
• Kummer theory

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 24
• Langue: English

Extrait

Jean-Pierre Wintenberger

Extensions of Iwasawa modules (2)

jw with Shekhar Khare

The aim of the talk is show how Leopoldt conjecture is linked with prop- erties of exact sequences of Iwasawa modules arising from ramification at auxiliary primes.The hope is to be able to use modular technics to study these properties. We restrict to the case of a totally realF. Letp >Let2 be a prime. F ∞ =F(µ p The cyclotomic character) be the cyclotomic extension.χ p ∞ ∗ identifies Gal(F ∞ /F) to an open subgroup of (Z p ) , hence the quotient by its torsion is isomorphic toZ p . Tothis quotient, corresponds the cyclotomic Z p -extension we callF ∞ . A formulation of Leopoldt conjecture (LC) is thatF ∞ is the onlyZ p - extension ofF. LetE F be the group of units ofF. LetU F be the units in thep-adic Q completion ofO F , soU F =U ℘ where the℘are the primes ofFover ℘ Q ∗ p. ThegroupU F (is the product the multiplicative groupsk ℘ ) ofthe ℘ Q 1 1 residue fileds by the groupUof units that reduces to 1 ink ℘ .Uis a F ℘ F Z p -module of rankrwherer= [F:Qhas an injection]. OneE F →U F . Let ¯ E F be the closure of the image ofE F inU F (with itsp-adic or congruence topology). LetL ∞ be the maximal abelianp-extension ofFthat is only ramified atpfield theory gives an exact sequence :. Class

¯ 1→U F /E F →Gal(L ∞ /F)→Cl(F)→1.

As aZ p -extension is unramified outsidep, it follows that LC is equivalent ¯ to the fact that theZ p rank ofE F isr−1,i.e.the rank ofE F . Itis equivalent to the fact that the topology onE F induced by the topology of U F is the same as thep-adic topology.On can express this by the fact that ap-adic regulator made fromE F and thep-adic logarithms is non zero.By thep-adic formula of Colmez for the residue of thep-adicL-functionζ F,p (s) ats= 1, it is equivalent to the fact thatζ p (s) has a pole ats= 1. RemarkLC is known for abelian extensions ofQand imaginary quadratic fields by Brumer using independence of logarithms technics and Galois prop- erties of units. One has another formulation by Iwasawa.Letqbe a primes ofFprime top. IfLis a finite abelianp-extension ofF, the inertia subroupI q (L/F) ∗ of Gal(L/F) is a quotient of (k q ) killedby a power ofp, hence it is cyclic of order divisible by thep-parte(q) ofN(q)−call1. IwasawaLfully ramified ifI q (L/F) has ordere(qproved that the Leopoldt conjecture). Iwasawa is true forFandpif and only if , for eachqprime ofFprime top,F 1