Shimura variety

Shimura variety
Name	Shimura variety
Type	Arithmetic variety
Field	Algebraic geometry; Number theory; Representation theory

Shimura variety is a class of higher-dimensional algebraic varieties arising from arithmetic quotients of hermitian symmetric domains by congruence subgroups of reductive algebraic groups, central to modern Number theory, Algebraic geometry, and Representation theory. These varieties generalize classical objects such as modular curves and carry rich structures linking Galois group actions, Hecke operator correspondences, and Hodge-theoretic period maps. Shimura varieties serve as geometric incarnations of instances of the Langlands program and provide testing grounds for conjectures by Deligne, Taniyama–Shimura, Mumford, and Kottwitz.

Definition and basic properties

A Shimura variety is defined from a pair (G,X) where G is a connected reductive algebraic group over Q and X is a G(R)-conjugacy class of homomorphisms from the Deligne torus to G_R satisfying Deligne's axioms; key formalizers include Deligne, Shimura, and Milne. For a neat compact open subgroup K⊂G(A_f) one forms the double quotient G(Q)\X×G(A_f)/K which, by work of Baily–Borel and Borel, acquires the structure of a quasi-projective algebraic variety defined over a number field called the reflex field; this construction uses results from Chevalley, Langlands, and Satake. Shimura varieties admit compatible systems of Hecke correspondences indexed by elements of G(A_f), and possess canonical models with actions of the absolute Galois group Gal(Qbar/ReflexField) studied by Shimura, Tate, and Serre.

Examples and classical cases

Basic examples include modular curves arising from G=GL(2) and X the upper half-plane, studied by Modular curves and Eichler, with congruence subgroups like Γ0(N) and Γ1(N). Hilbert modular varieties come from Res_{F/Q}GL(2) for a totally real field F and were treated by Hilbert, Shimura, and Matsumoto. Siegel modular varieties correspond to G=GSp(2g) and parametrize principally polarized abelian varieties, central to work by Siegel, Igusa, and Mumford. Unitary Shimura varieties arise from unitary groups attached to CM extensions and feature in research by Kudla, Rapoport, and Zink. Orthogonal Shimura varieties relate to quadratic forms and lattices studied by Kneser, Borcherds, and Gross–Zagier contexts.

Moduli interpretation and points over fields

Many Shimura varieties admit moduli interpretations as parameter spaces for abelian varieties with extra structure: polarizations, endomorphisms and level structure, following approaches by Mumford, Pappas, and Rapoport. For example, Siegel varieties parametrize principally polarized abelian varieties with symplectic level structures, linking to Tate module theory and Serre–Tate theory for p-adic deformation. Points over finite fields connect to Honda–Tate theory and the classification of isogeny classes by Frobenius endomorphisms as in work by Honda and Tate. Canonical models over reflex fields allow studying complex multiplication points tied to Shimura–Taniyama reciprocity and class field theory developed by Kronecker and Weber.

Arithmetic and automorphic aspects

Shimura varieties are arithmetic avatars of automorphic representations: the L^2-spectrum on G(Q)\G(A) relates via the Arthur–Selberg trace formula and work of Arthur, Langlands, Kottwitz, and Moy–Prasad to the étale cohomology of Shimura varieties. Hecke eigenclasses produce Galois representations as conjectured in the Langlands correspondence and proven in many cases by Taylor, Harris, Clozel, and Harris–Taylor. The models at primes of bad reduction involve integral models and local models studied by Rapoport–Zink, Kisin, and Pappas–Zhu, while the study of supersingular loci engages Oort and Li–Oort. Relations to special values of L-functions use results by Gross, Zagier, Kolyvagin, and Waldspurger.

Geometry: compactifications and cohomology

Compactifications of Shimura varieties were constructed by Baily–Borel (Satake–Baily–Borel) and refined by toroidal compactifications developed by Mumford, Ash, and Faltings–Chai. Intersection cohomology and perverse sheaf techniques applied by Goresky–MacPherson and Zucker illuminate Hodge theory on singular compactifications; Zucker's conjecture and its resolution by Looijenga and Saper–Stern connect L^2-cohomology to intersection cohomology. The mixed Hodge structures on cohomology relate to works by Deligne and Beilinson, and the contribution of endoscopic and stable trace formula methods by Labesse and Ngô impacts the calculation of cohomological automorphic multiplicities.

Applications and conjectures

Shimura varieties underpin major conjectures and results: they play roles in the proof of the Modularity theorem for elliptic curves by Wiles and Taylor–Wiles, inform the André–Oort conjecture studied by Pila, Tsimerman, and Tsimerman–Pila methods, and appear in the formulation of the Arithmetic Gan–Gross–Prasad conjecture advanced by Gan, Gross, and Prasad. They feature in constructions of Euler systems by Kato and Kolyvagin and in the study of rational points via Faltings and Mordell-type finiteness results. Open problems include general cases of the Langlands reciprocity, the full description of Newton strata in characteristic p per Mantovan and Shin, and the precise relationship between special cycles and derivatives of L-functions as conjectured by Kudla and Rapoport.

Category:Algebraic varieties Category:Number theory Category:Representation theory