Shimura variety

In mathematics, a Shimura variety is an analogue of a modular curve, and is (roughly) a quotient of an Hermitian symmetric space by a congruence subgroup of an algebraic group. The simplest example is the quotient of the upper half-plane by SL₂(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 by compactification of selected quotients of that type, were introduced in a series of papers of Goro Shimura during the 1960s. Shimura's approach was largely phenomenological, pursuing the widest generalizations of the reciprocity law formulation of complex 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 of modular curve. Abstract characterizations were introduced, to the effect that Shimura varieties are parameter spaces of certain types of Hodge structures. In other words the moduli problems 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 algebraic field 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 to Robert Langlands, the Hasse-Weil zeta function of any algebraic variety X defined over a number field K should be an automorphic L-function, i.e. should arise from an automorphic 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 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.

Michael Harris. Shimura varieties and the search for a Langlands transform

References

* James Arthur (Editor), David Ellwood (Editor), Robert Kottwitz (Editor) 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 MR0654325

* 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. MR0546620

* 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. MR0498581

* Milne, J.S. (2001), "Shimura variety", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104

* 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 (2001), "Shimura variety", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104

* J. S. Milne 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