formulasearchengine - User contributions [en]

Equations of motion

2014-02-27T21:58:54Z

108.20.254.43: /* Newton's second law for rotation */

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List of mathematical series

2014-02-14T12:30:12Z

108.20.32.196: /* Binomial coefficients */

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Total peripheral flow resistance

2013-03-04T19:22:07Z

108.20.168.66:

In mathematics, the '''Lubin–Tate formal group law''' is a [[formal group law]] introduced by {{harvs|txt|last=Lubin|author1-link=Jonathan Lubin|last2=Tate|author2-link=John Tate|year=1965}} to isolate the [[local field]] part of the classical theory of [[complex multiplication]] of [[elliptic function]]s. It does this by considering the (formal) [[endomorphism]]s of the formal group, emulating the way in which [[elliptic curve]]s with extra endomorphisms are used to give abelian extensions of [[global field]]s

==Definition of formal groups==

Let '''Z'''''p'' be the ring of ''p''-adic integers. The '''Lubin–Tate formal group law''' is the unique (1-dimensional) formal group law ''F'' such that ''e''(''x'') = ''px'' + ''x''''p'' is an endomorphism of ''F'', in other words

:<math>e(F(x,y)) = F(e(x), e(y)).\ </math>

More generally, the choice for ''e'' may be any power series such that

:''e''(''x'') = ''px'' + higher-degree terms and

:''e''(''x'') = ''x''''p'' mod ''p''.

All such group laws, for different choices of ''e'' satisfying these conditions, are strictly isomorphic.<ref>{{cite book | first1=Yu. I. | last1=Manin | authorlink1=Yuri I. Manin | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 | page=168 }}</ref> We choose these conditions so as to ensure that they reduce modulo the maximal ideal to Frobenius and the derivative at the origin is the [[Discrete valuation ring#Uniformizing parameter|prime element]].

For each element ''a'' in '''Z'''''p'' there is a unique endomorphism ''f'' of the Lubin–Tate formal group law such that ''f''(''x'') = ''ax'' + higher-degree terms. This gives an action of the ring '''Z'''''p'' on the Lubin–Tate formal group law.

There is a similar construction with '''Z'''''p'' replaced by any complete [[discrete valuation ring]] with finite [[Residue field of a valuation|residue class field]], where ''p'' is replaced by a choice of [[Uniformising parameter|uniformizer]].<ref name=K6263>{{cite book | first=Helmut | last=Koch | title=Algebraic Number Theory | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-63003-1 | zbl=0819.11044 | series=Encycl. Math. Sci. | volume=62 | edition=2nd printing of 1st | pages=62–63 }}</ref>

==Example==
We outline here a formal group equivalent of the [[Frobenius endomorphism|Frobenius element]], which is of great importance in [[class field theory]],<ref>e.g. Serre (1967). {{cite journal | last=Hazewinkel | first=Michiel | author-link = Michiel Hazewinkel | title=Local class field theory is easy | journal=Advances in Math. | volume=18 | year=1975 | issue=2 | pages=148–181 | zbl=0312.12022 }}</ref> generating the [[Hilbert class field|maximal unramified extension]] as the image of the reciprocity map.

For this example we need the notion of an endomorphism of formal groups, which is a formal group homomorphism ''f'' where the domain is the codomain. A formal group homomorphism from a formal group ''F'' to a formal group ''G'' is a power series over the same ring as the formal groups which has zero constant term and is such that:

:<math>f(F(X,Y))=G(f(X),f(Y))</math>

Consider a formal group ''F(X,Y)'' with coefficients in the ring of integers in a local field (for example '''Z'''''p''), taking ''X'' and ''Y'' to be in the unique maximal ideal gives us a convergent power series and in this case we define ''F(X,Y)'' = ''X'' +''F'' ''Y'' and we have a genuine group law. For example if ''F(X,Y)=X+Y'', then this is the usual addition. This is isomorphic to the case of ''F(X,Y)=X+Y+XY'', where we have multiplication on the set of elements which can be written as 1 added to an element of the prime ideal. In the latter case ''f(S)'' = (''I + S'')''p''-1 is an endomorphism of F and the isomorphism identifies f with the Frobenius element.

==Generating ramified extensions==
Lubin–Tate theory is important in explicit [[local class field theory]]. The [[unramified extension|unramified part]] of any abelian extension is easily constructed, Lubin–Tate finds its value in producing the ramified part. This works by defining a family of modules (indexed by the natural numbers) over the ring of integers consisting of what can be considered as roots of the power series repeatedly composed with itself. The compositum of all fields formed by adjoining such modules to the original field gives the ramified part.

A '''Lubin–Tate extension''' of a local field ''K'' is an abelian extension of ''K'' obtained by considering the ''p''-division points of a Lubin–Tate group. If ''g'' is an [[Eisenstein polynomial]], ''f''(''t'') = ''t'' ''g''(''t'') and ''F'' the Lubin–Tate formal group, let θ''n'' denote a root of ''gf''''n''-1(''t'')=''g''(''f''(''f''(⋯(''f''(''t''))⋯))). Then ''K''(θ''n'') is an abelian extension of ''K'' with Galois group isomorphic to ''U''/1+'''p'''''n'' where ''U'' is the unit group of the ring of integers of ''K'' and '''p''' is the maximal ideal.<ref name=K6263/>

==Connection with stable homotopy theory==

Lubin and Tate studied the [[deformation theory]] of such formal groups. A later application of the theory has been in the field of [[stable homotopy theory]], with the construction of a particular [[extraordinary cohomology theory]] associated to the construction for a given prime ''p''. As part of general machinery for formal groups, a cohomology theory with [[spectrum (homotopy theory)|spectrum]] is set up for the Lubin–Tate formal group, which also goes by the names of ''Morava E-theory'' or completed [[Johnson–Wilson theory]].<ref>http://www.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf</ref>

==References==
{{reflist}}

==Further reading ==
*{{citation | zbl=0674.12004 | last=de Shalit | first=Ehud | title=Iwasawa theory of elliptic curves with complex multiplication. p-adic L functions| series=Perspectives in Mathematics | volume=3 | publisher=Academic Press | year=1987 | isbn=0-12-210255-X }}
*{{Citation | last1=Iwasawa | first1=Kenkichi | authorlink=Kenkichi Iwasawa | title=Local class field theory | url=http://books.google.com/books?id=iJ7vAAAAMAAJ | publisher=The Clarendon Press Oxford University Press | series=Oxford Mathematical Monographs | isbn=978-0-19-504030-2 | mr=863740 | year=1986 | zbl=0604.12014 }}
*{{Citation | last1=Lubin | first1=Jonathan | last2=Tate | first2=John | author2-link=John Tate | title=Formal complex multiplication in local fields | jstor=1970622 | mr=0172878 | year=1965 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=81 | pages=380–387 | zbl=0128.26501 }}
*{{Citation | last1=Lubin | first1=Jonathan | last2=Tate | first2=John | author2-link=John Tate | title=Formal moduli for one-parameter formal Lie groups | url=http://www.numdam.org/item?id=BSMF_1966__94__49_0 | mr=0238854 | year=1966 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=94 | pages=49–59 | zbl=0156.04105 }}
*{{Neukirch ANT}}
*{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | editor1-first=John William Scott | editor2-last=Frohlich | title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) | editor1-first=J.W.S. | editor1-last=Cassels | editor1-link=J. W. S. Cassels | editor2-first=Albrecht | editor2-last=Fröhlich | editor2-link=Albrecht Fröhlich | year=1967 | publisher=Academic Press | zbl=0153.07403 | mr=0220701 | chapter=Local class field theory | pages=128–161}}

==External links==
*{{citation|url=http://www.math.harvard.edu/~lurie/252xnotes/Lecture21.pdf|title=Lubin–Tate theory|last=Lurie|first=J.|year=2010}}

{{DEFAULTSORT:Lubin-Tate formal group law}}
[[Category:Algebraic number theory]]

Sołtan argument

2012-03-15T07:03:01Z

108.20.106.214: /* History */ Minor correction concerning coinage of term 'black hole'. Reference info contained in article entry "black hole", already linked in the corrected sentence.

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