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In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. The term "Shimura variety" applies to the higher-dimensional case, in the case of one-dimensional varieties one speaks of Shimura curves. Hilbert modular surfaces and Siegel modular varieties are among the best known classes of Shimura varieties.

Special instances of Shimura varieties were originally introduced by Goro Shimura in the course of his generalization of the complex multiplication theory. Shimura showed that while initially defined analytically, they are arithmetic objects, in the sense that they admit models defined over a number field, the reflex field of the Shimura variety. In the 1970s, Pierre Deligne created an axiomatic framework for the work of Shimura. Around the same time Robert Langlands remarked that Shimura varieties form a natural realm of examples for which equivalence between motivic and automorphic L-functions postulated in the Langlands program can be tested. Automorphic forms realized in the cohomology of a Shimura variety are more amenable to study than general automorphic forms; in particular, there is a construction attaching Galois representations to them.

Definition

Shimura datum

Let S = Res_C/R G_m be the Weil restriction of the multiplicative group from complex numbers to real numbers. It is a real algebraic group, whose group of R-points, S(R), is C^* and group of C-points is C^*×C^*. A Shimura datum is a pair (G, X) consisting of a reductive algebraic group G defined over the field Q of rational numbers and a G(R)-conjugacy class X of homomorphisms h: S → G_R satisfying the following axioms:

• For any h in X, only weights (0,0), (1,−1), (−1,1) may occur in g_C, i.e. the complexified Lie algebra of G decomposes into a direct sum

$\mathfrak{g}\otimes \mathbb{ℂ}=\mathfrak{k}\oplus {\mathfrak{p}}^{+}\oplus {\mathfrak{p}}^{-},$

where for any z ∈ S, h(z) acts trivially on the first summand and via $z/\overline{z}$ (respectively, $\overline{z}/z$) on the second (respectively, third) summand.

• The adjoint action of h(i) induces a Cartan involution on the adjoint group of G_R.

• The adjoint group of G_R does not admit a factor H defined over Q such that the projection of h on H is trivial.

It follows from these axioms that X has a unique structure of a complex manifold (possibly, disconnected) such that for every representation ρ: G_R → GL(V), the family (V, ρ ⋅ h) is a holomorphic family of Hodge structures; moreover, it forms a variation of Hodge structure, and X is a finite disjoint union of hermitian symmetric domains.

Shimura variety

Let A_ƒ be the ring of adeles of Q. For any sufficiently small compact open subgroup K of G(A_ƒ), the double coset space

$S{h}_K(G,X)=G(\mathbb{\mathbb{Q}})\mathrm{\setminus }X\times G({\mathbb{𝔸}}_f)/K$

is a finite disjoint union of locally symmetric varieties of the form Γ \ X⁺, where the plus superscript indicates a connected component. The varieties Sh_K(G,X) are complex algebraic varieties and they form an inverse system over all sufficiently small compact open subgroups K. This inverse system

$(S{h}_K(G,X){)}_K$

admits a natural right action of G(A_ƒ). It is called the Shimura variety associated with the Shimura datum (G, X) and denoted Sh(G, X).

History

For special types of hermitian symmetric domains and congruence subgroups Γ, algebraic varieties of the form Γ \ X = Sh_K(G,X) and their compactifications were introduced in a series of papers of Goro Shimura during the 1960s. Shimura's approach, later presented in his monograph, was largely phenomenological, pursuing the widest generalizations of the reciprocity law formulation of complex multiplication theory. In retrospect, the name "Shimura variety" was introduced by Deligne, who proceeded to isolate the abstract features that played role in Shimura's theory. In Deligne's formulation, Shimura varieties are parameter spaces of certain types of Hodge structures. Thus they form a natural higher-dimensional generalization of modular curves viewed as moduli spaces of elliptic curves with level structure. In many cases, the moduli problems to which Shimura varieties are solutions have been likewise identified.

Examples

Let F be a totally real number field and D a quaternion division algebra over F. The multiplicative group D^× gives rise to a canonical Shimura variety. Its dimension d is the number of infinite places over which D splits. In particular, if d = 1 (for example, if F = Q and D ⊗ R ≅ M₂(R)), fixing a sufficiently small arithmetic subgroup of D^×, one gets a Shimura curve, and curves arising from this construction are already compact (i.e. projective).

Some examples of Shimura curves with explicitly known equations are given by the Hurwitz curves of low genus:

• Klein quartic (genus 3)
• Macbeath surface (genus 7)
• First Hurwitz triplet (genus 14)

and by the Fermat curve of degree 7.

Other examples of Shimura varieties include Picard modular surfaces and Hilbert–Blumenthal varieties.

Canonical models and special points

Each Shimura variety can be defined over a canonical number field E called the reflex field. This important result due to Shimura shows that Shimura varieties, which a priori are only complex manifolds, have an algebraic field of definition and, therefore, arithmetical significance. It forms the starting point in his formulation of the reciprocity law, where an important role is played by certain arithmetically defined special points.

The qualitative nature of the Zariski closure of sets of special points on a Shimura variety is described by the André-Oort conjecture. Conditional results have been obtained on this conjecture, assuming a Generalized Riemann Hypothesis.

Role in the Langlands program

Shimura varieties play an outstanding role in the Langlands program. The prototypical theorem, the Eichler–Shimura congruence relation, implies that the Hasse-Weil zeta function of a modular curve is a product of L-functions associated to explicitly determined modular forms of weight 2. Indeed, it was in the process of generalization of this theorem that Goro Shimura introduced his varieties and proved his reciprocity law. Zeta functions of Shimura varieties associated with the group GL₂ over other number fields and its inner forms (i.e. multiplicative groups of quaternion algebras) were studied by Eichler, Shimura, Kuga, Sato, and Ihara. On the basis of their results, Robert Langlands made a prediction that the Hasse-Weil zeta function of any algebraic variety W defined over a number field would be a product of positive and negative powers of automorphic L-functions, i.e. it should arise from a collection of automorphic representations. However philosophically natural it may be to expect such a description, statements of this type have only been proved when W is a Shimura variety. In the words of Langlands:

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References

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  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
• James Arthur, David Ellwood, and Robert Kottwitz (ed) Harmonic Analysis, the Trace Formula and Shimura Varieties, Clay Mathematics Proceedings, vol 4, AMS, 2005 ISBN 978-0-8218-3844-0
• Pierre Deligne, Travaux de Shimura. Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, pp. 123–165. Lecture Notes in Math., Vol. 244, Springer, Berlin, 1971. Template:MathSciNet, Numdam
• Pierre Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, in Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math., XXXIII (Corvallis, OR, 1977), Part 2, pp. 247–289, Amer. Math. Soc., Providence, R.I., 1979. Template:MathSciNet
• Pierre Deligne, James S. Milne, Arthur Ogus, Kuang-yen Shi, Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, 900. Springer-Verlag, Berlin-New York, 1982. ii+414 pp. ISBN 3-540-11174-3 Template:MathSciNet
• 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.
  
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• J. Milne, Shimura varieties and motives, in U. Jannsen, S. Kleiman. J.-P. Serre (ed.), Motives, Proc. Symp. Pure Math, 55:2, Amer. Math. Soc. (1994), pp. 447–523
• J. S. Milne, Introduction to Shimura varieties, in Arthur, Ellwood, and Kottwitz (2005)
• Harry Reimann, The semi-simple zeta function of quaternionic Shimura varieties, Lecture Notes in Mathematics, 1657, Springer, 1997
• Goro Shimura, The Collected Works of Goro Shimura (2003), vol 1–5
• Goro Shimura Introduction to Arithmetic Theory of Automorphic Functions