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In [[algebraic geometry]], the '''Mumford–Tate group''' (or '''Hodge group''') ''MT''(''F'') constructed from a [[Hodge structure]] ''F'' is a certain [[algebraic group]] ''G''. When ''F'' is given by a [[rational representation]] of an [[algebraic torus]], the definition of ''G'' is as the [[Zariski closure]] of the image in the representation of the [[circle group]], over the [[rational numbers]]. {{harvs|txt|authorlink=David Mumford|last=Mumford|year=1966}} introduced Mumford–Tate groups over the complex numbers under the name of Hodge groups. {{harvtxt|Serre|1967}} introduced the ''p''-adic analogue of Mumford's construction for [[Hodge–Tate module]]s, using the work of {{harvs|txt|last=Tate|authorlink=John Tate|year=1967}} on [[p-divisible group]]s, and named them Mumford–Tate groups. | |||
==Formulation== | |||
The algebraic torus ''T'' used to describe Hodge structures has a concrete matrix representation, as the 2×2 invertible matrices of the shape that is given by the action of ''a''+''bi'' on the basis {1,''i''} of the complex numbers '''C''' over '''R''': | |||
:<math>\begin{bmatrix} a & b \\ -b & a \end{bmatrix}. | |||
</math> | |||
The circle group inside this group of matrices is the [[unitary group]] ''U''(1). | |||
Hodge structures arising in geometry, for example on the [[cohomology group]]s of [[Kähler manifold]]s, have a [[lattice (group theory)|lattice]] consisting of the integral cohomology classes. Not quite so much is needed for the definition of the Mumford–Tate group, but it does assume that the vector space ''V'' underlying the Hodge structure has a given rational structure, i.e. is given over the rational numbers ''Q''. For the purposes of the theory the complex vector space ''V''<sub>''C''</sub>, obtained by extending the scalars of ''V'' from ''Q'' to ''C'', is used. | |||
The weight ''k'' of the Hodge structure describes the action of the diagonal matrices of ''T'', and ''V'' is supposed therefore to be homogeneous of weight ''k'', under that action. Under the action of the full group ''V''<sub>''C''</sub> breaks up into subspaces ''V''<sup>''pq''</sup>, complex conjugate in pairs under switching ''p'' and ''q''. Thinking of the matrix in terms of the complex number λ it represents, ''V''<sup>''pq''</sup> has the action of λ by the ''p''th power and of the complex conjugate of λ by the ''q''th power. Here necessarily | |||
:''p'' + ''q'' = ''k''. | |||
In more abstract terms, the torus ''T'' underlying the matrix group is the [[Weil restriction]] of the [[multiplicative group]] ''GL''(1), from the complex field to the real field, an algebraic torus whose character group consists of the two homomorphisms to ''GL''(1), interchanged by complex conjugation. | |||
Once formulated in this fashion, the rational representation ρ of ''T'' on ''V'' setting up the Hodge structure ''F'' determines the image ρ(''U''(1)) in ''GL''(''V''<sub>''C''</sub>); and ''MT''(''F'') is by definition the Zariski closure, for the ''Q''-Zariski topology on ''GL''(''V''), of this image.<ref>http://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf, pp. 7–9.</ref> | |||
==Mumford–Tate conjecture== | |||
The original context for the formulation of the group in question was the question of the [[Galois representation]] on the [[Tate module]] of an [[abelian variety]] ''A''. Conjecturally, the image of such a Galois representation, which is an [[l-adic]] Lie group for a given prime number ''l'', is determined by the corresponding Mumford–Tate group ''G'' (coming from the Hodge structure on ''H''<sup>1</sup>(''A'')), to the extent that knowledge of ''G'' determines the [[Lie algebra]] of the Galois image. This conjecture is known only in particular cases.<ref>http://math.berkeley.edu/~ribet/Articles/mg.pdf, survey by [[Ken Ribet]].</ref> Through generalisations of this conjecture, the Mumford–Tate group has been connected to the [[motivic Galois group]], and, for example, the general issue of extending the [[Sato–Tate conjecture]] (now a theorem). | |||
==Period conjecture== | |||
A related conjecture on abelian varieties states that the [[period matrix]] of ''A'' over number field has [[transcendence degree]], in the sense of the field generated by its entries, predicted by the dimension of its Mumford–Tate group, as in the previous section. Work of [[Pierre Deligne]] has shown that the dimension bounds the transcendence degree; so that the Mumford–Tate group catches sufficiently many algebraic relations between the periods. This is a special case of the full Grothendieck period conjecture.<ref>http://math.cts.nthu.edu.tw/Mathematics/preprints/prep2005-6-002.pdf, p. 3.</ref><ref>http://arxiv.org/abs/0805.2569v1, p. 7.</ref> | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
*{{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) | publisher=[[American Mathematical Society]] | location=Providence, R.I. | id={{MR|0206003}} | year=1966 | chapter=Families of abelian varieties | pages=347–351}} | |||
*{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | editor1-last=Springer | editor1-first=Tonny A. | title=Proceedings of a Conference on Local Fields (Driebergen, 1966) | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-03953-2 | id={{MR|0242839}} | year=1967 | chapter=Sur les groupes de Galois attachés aux groupes p-divisibles | pages=118–131}} | |||
*{{Citation | last1=Tate | first1=John T. | editor1-last=Springer | editor1-first=Tonny A. | title=Proc. Conf. Local Fields( Driebergen, 1966) | publisher=[[Springer-Verlag]] | location=Berlin, New York | mr=0231827 | year=1967 | chapter=p-divisible groups.}} | |||
==External links== | |||
*[http://www.math.umass.edu/~hacking/agnes/griffiths.pdf Lecture slides (PDF)] by [[Phillip Griffiths]] | |||
*[http://www.cms.zju.edu.cn/UploadFiles/AttachFiles/2005730145725232.pdf ''Mumford-Tate groups, families of Calabi-Yau varieties and analogue André-Oort problems I'', preprint (PDF)] | |||
{{DEFAULTSORT:Mumford-Tate group}} | |||
[[Category:Hodge theory]] | |||
[[Category:Algebraic groups]] | |||
Revision as of 23:12, 2 December 2012
In algebraic geometry, the Mumford–Tate group (or Hodge group) MT(F) constructed from a Hodge structure F is a certain algebraic group G. When F is given by a rational representation of an algebraic torus, the definition of G is as the Zariski closure of the image in the representation of the circle group, over the rational numbers. Template:Harvs introduced Mumford–Tate groups over the complex numbers under the name of Hodge groups. Template:Harvtxt introduced the p-adic analogue of Mumford's construction for Hodge–Tate modules, using the work of Template:Harvs on p-divisible groups, and named them Mumford–Tate groups.
Formulation
The algebraic torus T used to describe Hodge structures has a concrete matrix representation, as the 2×2 invertible matrices of the shape that is given by the action of a+bi on the basis {1,i} of the complex numbers C over R:
The circle group inside this group of matrices is the unitary group U(1).
Hodge structures arising in geometry, for example on the cohomology groups of Kähler manifolds, have a lattice consisting of the integral cohomology classes. Not quite so much is needed for the definition of the Mumford–Tate group, but it does assume that the vector space V underlying the Hodge structure has a given rational structure, i.e. is given over the rational numbers Q. For the purposes of the theory the complex vector space VC, obtained by extending the scalars of V from Q to C, is used.
The weight k of the Hodge structure describes the action of the diagonal matrices of T, and V is supposed therefore to be homogeneous of weight k, under that action. Under the action of the full group VC breaks up into subspaces Vpq, complex conjugate in pairs under switching p and q. Thinking of the matrix in terms of the complex number λ it represents, Vpq has the action of λ by the pth power and of the complex conjugate of λ by the qth power. Here necessarily
- p + q = k.
In more abstract terms, the torus T underlying the matrix group is the Weil restriction of the multiplicative group GL(1), from the complex field to the real field, an algebraic torus whose character group consists of the two homomorphisms to GL(1), interchanged by complex conjugation.
Once formulated in this fashion, the rational representation ρ of T on V setting up the Hodge structure F determines the image ρ(U(1)) in GL(VC); and MT(F) is by definition the Zariski closure, for the Q-Zariski topology on GL(V), of this image.[1]
Mumford–Tate conjecture
The original context for the formulation of the group in question was the question of the Galois representation on the Tate module of an abelian variety A. Conjecturally, the image of such a Galois representation, which is an l-adic Lie group for a given prime number l, is determined by the corresponding Mumford–Tate group G (coming from the Hodge structure on H1(A)), to the extent that knowledge of G determines the Lie algebra of the Galois image. This conjecture is known only in particular cases.[2] Through generalisations of this conjecture, the Mumford–Tate group has been connected to the motivic Galois group, and, for example, the general issue of extending the Sato–Tate conjecture (now a theorem).
Period conjecture
A related conjecture on abelian varieties states that the period matrix of A over number field has transcendence degree, in the sense of the field generated by its entries, predicted by the dimension of its Mumford–Tate group, as in the previous section. Work of Pierre Deligne has shown that the dimension bounds the transcendence degree; so that the Mumford–Tate group catches sufficiently many algebraic relations between the periods. This is a special case of the full Grothendieck period conjecture.[3][4]
Notes
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References
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010