- Shimura variety
In mathematics, a Shimura variety is an analogue of a
modular curve , and is (roughly) a quotient of anHermitian symmetric space by acongruence subgroup of analgebraic group . The simplest example is the quotient of theupper half-plane by "SL"2(Z). The term "Shimura variety" is usually reserved for the higher-dimensional case, in the case of one-dimensional varieties one speaks of Shimura curves.Such
algebraic varieties , formed bycompactification of selected quotients of that type, were introduced in a series of papers ofGoro Shimura during the 1960s. Shimura's approach was largely phenomenological, pursuing the widest generalizations of the reciprocity law formulation ofcomplex multiplication theory, in his book (see references). In retrospect, the name "Shimura variety" was introduced, to recognise that these varieties form the appropriate higher-dimensional class of complex manifolds building on the idea ofmodular curve . Abstract characterizations were introduced, to the effect that Shimura varieties are parameter spaces of certain types ofHodge structure s. In other words themoduli problem s to which Shimura varieties are solutions have been identified.Canonical models
The basic result applying to the required compactification is the
Baily-Borel theorem .Each Shimura variety can be defined over a canonical
number field "K". This important result due to Shimura shows that Shimura varieties, which "a priori" are complex manifolds, have an algebraicfield of definition and therefore arithmetical significance. It forms the starting point in his formulation of the reciprocity law.Role in the Langlands program
Shimura varieties play an important role in the
Langlands program . According toRobert Langlands , theHasse-Weil zeta function of any algebraic variety "X" defined over a number field "K" should be anautomorphic L-function , i.e. should arise from anautomorphic representation . However natural it may be to expect this, statements of this type have only been proved when "X" is a Shimura variety. (Qualification: many examples are known, and the sense in which they all "come from" Shimura varieties is a somewhat abstract one.)The prototypical theorem is given by the
Eichler-Shimura congruence relation , which implies that the Hasse-Weil zeta function of a modular curve is a product of L-functions associated to explicitly determinedmodular form s 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.References
*James Arthur (Editor), David Ellwood (Editor), Robert Kottwitz (Editor) [http://www.claymath.org/publications/Harmonic_Analysis/ "Harmonic Analysis, the Trace Formula, and Shimura Varieties"] : Proceedings of the Clay Mathematics Institute, 2003 Summer School, the Fields Institute, (Clay Mathematics Proceedings,) ISBN 082183844X
*Deligne, Pierre; Milne, James S.; Ogus, Arthur; Shih, Kuang-yen "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 MathSciNet|id=0654325
*Deligne, Pierre "Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques." Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 247--289, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979. MathSciNet|id=0546620
* Deligne, Pierre "Travaux de Shimura." Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, pp. 123--165. Lecture Notes in Math., Vol. 244, Springer, Berlin, 1971. MathSciNet|id=0498581
*springer|id=s/s110090|first=J.S. |last=Milne
*J. Milne, "Shimura varieties and motives" U. Jannsen (ed.) S. Kleiman (ed.) J.-P. Serre (ed.) , Motives , Proc. Symp. Pure Math. , 55: 2 , Amer. Math. Soc. (1994) pp. 447–523
*J. S. Milne [http://www.claymath.org/publications/Harmonic_Analysis/chapter2.pdf Introduction to Shimura varieties] , chapter 2 of the book edited by Arthur, Ellwood, and Kottwitz (2003)
*Shimura, Goro, "The Collected Works of Goro Shimura" (2003), five volumes
*Shimura, Goro "Introduction to Arithmetic Theory of Automorphic Functions"
Wikimedia Foundation. 2010.
