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A Shimura Variety? WHATIS... ? a Shimura Variety? James S. Milne Most mathematicians have encountered modular congruence subgroup functions. For example, when the group theorists def= f 2 j ≡ g discovered the monster group, they were surprised Ð (N) A SL2(Z) A I mod N . to find that the degrees of its irreducible represen- In this case, the Riemann surface Ð nD1 can be tations were already encoded in the q-coefficients compactified in a natural way by adding a finite of the j-function. The theory of Shimura varieties number of points (called the cusps), and so Ð nD1 grew out of the applications of modular functions has a unique structure of an algebraic curve and modular forms to number theory. Roughly compatible with its analytic structure. In all other speaking, Shimura varieties are the varieties on cases, Ð nD1 is compact and so is automatically an which modular functions live. algebraic curve. Each Shimura curve has a natural embedding in Shimura Curves projective space. Consider, for example, the elliptic According to the uniformization theorem, every def modular curve Y (N) = Ð (N)nD1. A cusp form of simply connected Riemann surface is isomorphic weight 2k for Ð (N) is a holomorphic function f on to the Riemann sphere, the complex plane, or the D1 vanishing at the cusps and such that open unit disk (equivalently the complex upper half (1) plane D1). The Shimura curves are the quotients of 2k a b f (Az) = (cz +d) ·f (z) for all A = c d 2 Ð (N). D1 by the actions of certain discrete groups, which I now describe. For some fixed k, a basis f0; : : : ; fn for the cusp a b az+b forms of weight 2k defines an embedding The action c d z = cz+d of SL2(R) on D1 realizes SL ( )/{±Ig as the group of holomor- n 2 R P , (f0(P): ::: : fn(P)): Y (N) ! P (C) phic automorphisms Hol(D1) of D1. Let B be a n quaternion algebra over a totally real number field of Y (N) as an algebraic subvariety of P (C). In fact, we can do better. Each of the cusps is fixed by F such that R ⊗F B is isomorphic to M2(R) for 1 h exactly one embedding of F into R, and let G a unipotent matrix 0 1 with h a positive integer. be the algebraic group over Q whose R-points For such a matrix, (1) becomes f (z) = f (z + h), for any Q-algebra R are the elements of B ⊗Q R and so a cusp form f has an expansion of norm 1. Then G(R) is a product of SL2(R) 2 3 2πi=h (2) f (z) = a1q + a2q + a3q + · · · ; q = e : with a compact group, and so there is a surjective homomorphism ' : G(R) ! Hol(D1) with compact Let Q[ζN ] be the field generated over Q by a kernel. A Shimura curve is the quotient of D1 by primitive Nth root of 1. It is possible to choose the image in Hol(D1) of a congruence subgroup of the basis f0; : : : ; fn so that the coefficients in (2) G(Q). lie in the field Q[ζN ] and, when this is done, the For example, when B = M2(Q), the group G is homogeneous polynomials defining Y (N) have SL2, and we get the familiar elliptic modular curves, coefficients in Q[ζN ]. Thus Y (N) has a canonical namely, the quotients of D1 by a discrete subgroup model over Q[ζN ]. This property of Y (N) is very unusual. Typically, an algebraic variety over C will Ð of Hol(D1) containing the image of a principal not have a model over an algebraic number field, James S. Milne is emeritus professor of mathematics at and when it does, it will have many distinct models, the University of Michigan. His email address is jmilne@ none of which is to be preferred. umich.edu. The above explanation for why Y (N) has a DOI: http://dx.doi.org/10.1090/noti923 canonical model over Q[ζN ] is that of the analysts. 1560 Notices of the AMS Volume 59, Number 11 The geometers have an entirely different explana- of the variety over a specific number field that tion. For an elliptic curve E over C, the group E(C)N is uniquely determined by specifying the fields of elements of order N is a free Z=NZ-module of generated by the coordinates of certain special rank 2 equipped with a skew-symmetric pairing eN points. Shimura and his students Miyake and taking values in the group of Nth roots of 1 in C.A Shih proved the existence of canonical models level-N structure on E is a basis (t1; t2) for E(C)N for several fundamental families of connected such that eN (t1; t2) = ζN . For N ≥ 3, the canonical Shimura varieties. model of Y (N) represents the functor sending a Q[ζN ]-algebra R to the set of isomorphism classes Shimura Varieties, according to Deligne of elliptic curves over R equipped with a level-N When Deligne was asked to report on Shimura’s structure. work in a 1971 Bourbaki seminar, he rewrote the Both explanations fail when B ≠ M2(Q): then foundations. For Deligne, a Shimura variety is the curves are compact, so there are no cusps and defined by a reductive group G over Q and a G(R)- no q-expansions, and they are not moduli varieties conjugacy class of homomorphisms h: C× ! G(R) in any natural way. Thus both the analysts and the satisfying certain axioms. The Shimura variety geometers were surprised when Shimura (in 1967) itself is a certain double coset space. The axioms proved that all these curves do have canonical ensure that, on the one hand, this double coset models over specific number fields. The theory of space is a finite disjoint union of the varieties Shimura varieties, as distinct from the theory of considered in the preceding section and on the moduli varieties, can be said to have been born other hand, that it is the base space for a variation with Shimura’s paper. Ihara (in 1968) attached of Hodge structures. Sometimes the variation of Shimura’s name to these curves. Hodge structures arises from a family of abelian varieties, in which case the existence of a canonical Shimura Varieties, according to Shimura model follows from the theory of moduli varieties. A complex manifold is symmetric if each point is an In other cases, Deligne was able to prove the isolated fixed point of an involution. For example, existence of a canonical model by relating the D1 is symmetric because it is homogeneous and Shimura variety to one that is a moduli variety. In i is an isolated fixed point of the involution the remaining cases, the existence of a canonical 0 −1 z , 1 0 z = −1=z. A connected symmetric model was proved by the author and Borovoi complex manifold is called a hermitian symmetric (somewhat independently). domain if it is isomorphic to a bounded open Shimura varieties interested Langlands as a subset of Cn for some n. Every hermitian symmetric source of Galois representations and as a test for domain is simply connected, and so the Riemann his idea that all zeta functions are automorphic. In mapping theorem shows that D1 is the only a 1974 lecture he introduced the term “Shimura hermitian symmetric domain of dimension one. variety” for the varieties defined by Deligne. Once The connected Shimura varieties are the quotients the existence (and uniqueness) of their canonical of hermitian symmetric domains by the actions of models had been demonstrated, it became cus- certain discrete groups, which I now describe. tomary to refer to the canonical model as the The group of holomorphic automorphisms of Shimura variety (rather than the variety over C). a hermitian symmetric domain D is a semisimple The connected components of these varieties are Lie group whose identity component we denote by the canonical models of the preceding section. Hol(D)+. To define a family of connected Shimura varieties covered by D, we need a semisimple Further Reading algebraic group G over Q and a surjective homo- For Shimura’s approach, I suggest looking first + morphism G(R) ! Hol(D) with compact kernel. at his notes Automorphic Functions and Number The connected Shimura varieties are then the Theory 1 and his ICM talks. For Deligne’s approach quotients Ð nD of D by a torsion-free subgroup Ð there are the difficult original articles of Deligne2 + of Hol(D) containing the image of a congruence and the author’s Introduction to Shimura Varieties.3 subgroup of G(Q) as a subgroup of finite index. Baily and Borel proved that, as in the curve case, the modular forms on D relative to Ð embed Ð nD as an algebraic subvariety of some projective space. Thus, each manifold Ð nD has a canonical structure as an algebraic variety over C, and a later theorem 1Lecture Notes in Mathematics, No. 54, Springer, 1968. of Borel shows that the algebraic structure is in 2Travaux de Shimura. Séminaire Bourbaki (1970/71), Exp. fact unique. No. 389; Variétés de Shimura, Proc. Sympos. Pure Math., 33, Shimura introduced the notion of a canonical AMS, 1979, pp. 247–289. model for these algebraic varieties. This is a model 3Clay Math. Proc., 4, AMS, 2005, pp. 265–378. December 2012 Notices of the AMS 1561.
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