Math 2.0

Algebraic geometry Definition

Shimura variety

Builds on Reductive groupAdele ringHodge structureSymmetric spaceQuasi-projective variety

Definition

Let π•Š=Res⁑ℂ/ℝ𝔾m\mathbb{S} = \operatorname{Res}_{\C/\R} \mathbb{G}_m be the Deligne torus, so that π•Š(ℝ)=β„‚Γ—\mathbb{S}(\R) = \C^\times. Let GG be a connected reductive group over β„š\Q and let XX be a G(ℝ)G(\R)-conjugacy class of homomorphisms h ⁣:π•Šβ†’Gℝh \colon \mathbb{S} \to G_\R. The pair (G,X)(G, X) is a Shimura datum if for every h∈Xh \in X:

  1. The Hodge structure on the Lie algebra of GℝG_\R defined by Ad⁑∘h\operatorname{Ad} \circ h has only the types (βˆ’1,1)(-1, 1), (0,0)(0, 0) and (1,βˆ’1)(1, -1).
  2. Conjugation by h(i)h(i) is a Cartan involution of the adjoint group GℝadG^{\mathrm{ad}}_\R.
  3. GadG^{\mathrm{ad}} has no factor defined over β„š\Q on which the projection of hh is trivial.

Let 𝔸f\mathbb{A}_f be the finite adeles of β„š\Q and let KβŠ†G(𝔸f)K \subseteq G(\mathbb{A}_f) be a compact open subgroup. The Shimura variety of level KK is Sh⁑K(G,X)=G(β„š)\(XΓ—G(𝔸f)/K),\operatorname{Sh}_K(G, X) = G(\Q) \backslash \big( X \times G(\mathbb{A}_f) / K \big), with G(β„š)G(\Q) acting diagonally and KK acting on the right of G(𝔸f)G(\mathbb{A}_f).

Remarks

The conditions make XX a finite union of Hermitian symmetric spaces, and Sh⁑K(G,X)\operatorname{Sh}_K(G, X) is a finite disjoint union of quotients Ξ“\X+\Gamma \backslash X^+ by arithmetic subgroups Ξ“\Gamma. By the theorem of Baily and Borel, it is a quasi-projective variety over β„‚\C, and it is smooth when KK is small enough. By work of Shimura, Deligne and others it has a canonical model over a number field, the reflex field.

Example: let G=GL2G = \mathrm{GL}_2 and let XX be the conjugacy class of h(a+bi)=(aβˆ’bba)h(a + bi) = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}. Then XX is identified with β„‚βˆ–β„\C \setminus \R, the union of the upper and lower half-planes, and the Shimura varieties Sh⁑K(GL2,X)\operatorname{Sh}_K(\mathrm{GL}_2, X) are finite unions of modular curves. For G=GSp2gG = \mathrm{GSp}_{2g} one obtains the Siegel modular varieties, which parametrize principally polarized abelian varieties of dimension gg with level structure.

Non-example: G=GL3G = \mathrm{GL}_3 admits no Shimura datum, because the symmetric space of SL3(ℝ)\mathrm{SL}_3(\R) has no invariant complex structure, so clauses 1 and 2 cannot both hold.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Adele ring, concept, no evidence levelAdele ringconceptHodge structure, concept, no evidence levelHodge structureconceptQuasi-projective variety, concept, no evidence levelQuasi-projective vari…conceptReductive group, concept, no evidence levelReductive groupconceptSymmetric space, concept, no evidence levelSymmetric spaceconceptShimura variety, definition, no evidence levelShimura varietydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Adele ring concept
  2. Hodge structure concept
  3. Quasi-projective variety concept
  4. Reductive group concept
  5. Symmetric space concept
  6. Shimura variety definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Shimura_variety
About this page
Tag53HPW59SJC
URIm2://53HPW59SJC
Kinddefinition
WritingW0 stub
MSC14G35, 11G18
DefinesShimura variety
UsesReductive group, Adele ring, Hodge structure, Symmetric space, Quasi-projective variety
Changed2 November 2026

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