Linear algebra over Zp u

Description

Linear algebra over Zp[[u]] Xavier Caruso, David Lubicz IRMAR — University Rennes 1 June 20, 2011

• module over


• adic galois


• irmar —


• iwasawa theory


• galois group


• ring equipped


• linear algebra over

Sujets

• Lubicz
• Caruso
• Iwasawa theory
• Galois group

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 7
• Langue: English

Extrait

Linear

algebra

over

Z p [[u]]

Xavier Caruso, David Lubicz

IRMAR — University Rennes 1

June 20, 2011

ezorroimdnesWfa,Letunifπbea riscefbe tahe De lkiA .] tfom W=St]u[[ ewithS-mtocomputogirhtsmcfieitnla

•for allx∈W,v W (x) = +∞iffx=0; •for allx,y∈W,v W (xy) =v W (x) +v W (y); •for allx,y∈W,v W (x+y)≥min(v W (x),v W (y)); •for allx∈W,v W (x) =0 iffxis invertible.

v W :W→N∪ {+∞}such that

that is a ring equipped with a surjective map

LetWbe a discrete valuation ring,

Notations

.seludo

talkftheAimo u]]. eitnfecfiirebeDcsmpcotomsthrigoalseludom-Shtiwetu Stes[[W=dna .

LetWbe a discrete valuation ring.

Notations

the ring ofp-adic integersZ p ;v W , here, is the usualp-adic valuation

Examples:

the ring of integers of a finite extension ofQ p

the ringk[[X]]wherekis a field;v W , here, is the usual valuation of a serie (that is the smallest power ofXhaving a nonzero coefficient)

Letπbe a uniformizer ofW, that is an element such thatv W (π) =1.

Notations

LetWbe a discrete valuation ring.

Examples:

the ring ofp-adic integersZ p ;v W , here, is the usualp-adic valuation

the ringk[[X]]wherekis a field;v W , here, is the usual valuation of a serie (that is the smallest power ofXhaving a nonzero coefficient)

the ring of integers of a finite extension ofQ p

Letπbe a uniformizer ofW, and setS=W[[u]].

Aim of the talk Describe efficient algorithms to compute withS-modules.

Motivations

Well, it is certainly interesting for itself, but concretely we expect applications to :

Iwasawa theory Certain abelian Galois groups inherit a structure of Z p [[u]]-module (and Iwasawa used this fact to study them).

p-adic Hodge theory ( ciseni eprsereant latts ti e on m s i-s o t f a G bl Q e p ) − ∼ → ( +ansltioirtu rsove add omeludZ c p t [[ u u r ] e ] s )

Example: (restrictions toG Q p of)p-adic Galois representations associated to a modular form of level prime topare semi-stable (and even crystalline).

Precise set-up

Recall that we want to describe efficient algorithms to manipulateS-modules,e.g.compute intersections, sums, kernels, images,etc.

Basic assumption: We restrict ourselves to finitely generated moduleswithout torsion.

All these modules can be realized as submodules ofS d for a suitabled. In the sequel, we will always assume that our modulesare embedded in someS d andhave full rank(i.e.contain a family ofdvectors linearly independant).

). In rank equethes osSfudelufll( d fo lusoteleotstontilliwew,lpmocevig precision ofcours niadogdoonitnofo tabuo,lsrsfomoub of,eelertnemSnis omrfboelrP F noisicerpfomelbro n P ioatntsereep roopemblfir he twstnihtofstna,ssujddone.

Preliminary problems

Theoretical problem

rthethir

S

is not a very nice ring (e.g.it is not a principal domain).

Example: for alln, the ideal(π n , π n−1 u, . . . ,u n

)cannot be gen-

erated by less thann+1 elements.