Linear algebra over Zp u
79 pages
English

Linear algebra over Zp u

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79 pages
English
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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


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Publié par
Nombre de lectures 7
Langue English

Exrait

Linear
algebra
over
Zp[[u]]
Xavier Caruso, David Lubicz
IRMAR — University Rennes 1
June 20, 2011
ezorroimdnesWfa,LetunifπbeariscefbetaheDelkiA.]tfomW=St]u[[ewithS-mtocomputogirhtsmceitnla
for allxW,vW(x) = +iffx=0; for allx,yW,vW(xy) =vW(x) +vW(y); for allx,yW,vW(x+y)min(vW(x),vW(y)); for allxW,vW(x) =0 iffxis invertible.
vW:WN∪ {+∞}such that
that is a ring equipped with a surjective map
LetWbe a discrete valuation ring,
Notations
.seludo
talkftheAimou]].eitnfecirebeDcsmpcotomsthrigoalseludom-ShtiwetuStes[[W=dna.
LetWbe a discrete valuation ring.
Notations
the ring ofp-adic integersZp;vW, here, is the usualp-adic valuation
Examples:
the ring of integers of a finite extension ofQp
the ringk[[X]]wherekis a field;vW, 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 thatvW(π) =1.
Notations
LetWbe a discrete valuation ring.
Examples:
the ring ofp-adic integersZp;vW, here, is the usualp-adic valuation
the ringk[[X]]wherekis a field;vW, 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 ofQp
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 Zp[[u]]-module (and Iwasawa used this fact to study them).
p-adic Hodge theory (cisenieprsereantlattstieonmsi-sotfaGblQep)(+ansltioirtursoveaddomeludZcpt[[uur]e]s)
Example: (restrictions toGQpof)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 ofSdfor a suitabled. In the sequel, we will always assume that our modulesare embedded in someSdandhave full rank(i.e.contain a family ofdvectors linearly independant).
).InrankequethesosSfudelufll(dfolusoteleotstontilliwew,lpmocevigprecisionofcoursniadogdoonitnofotabuo,lsrsfomoubof,eelertnemSnisomrfboelrPFnoisicerpfomelbronPioatntsereeproopemblrhetwstnihtofstna,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(πn, πn1u, . . . ,un
)cannot be gen-
erated by less thann+1 elements.
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