nLab Shimura variety




A Shimura variety is a higher-dimensional analog of a modular curve.


A Shimura datum is a pair (G,X)(G,X) where GG is a reductive group and XX is a G()G(\mathbb{R})-conjugacy class of homorphisms h:𝕊G h:\mathbb{S}\to G_{\mathbb{R}}, where 𝕊=Res /𝔾 m\mathbb{S}=\Res_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m}.

Given a Shimura datum G(X)G(X) and a sufficiently small compact open subgroup KK of G(𝔸 f)G(\mathbb{A}_{f}) (where 𝔸 f= p\mathbb{A}_{f}=\prod'\mathbb{Q}_{p} is the finite adeles), the quotient

Sh K(G,X):=G()\G(𝔸 f)×X/K\Sh_{K}(G,X):=G(\mathbb{Q})\backslash G(\mathbb{A}_{f})\times X/K

can be realized as a quasi-projective variety over \mathbb{C} (and actually over a canonical number field called the reflex field that depends on the Shimura datum). The inverse limit Sh(G,X):=(Sh K(G,X)) K\Sh(G,X):=(\Sh_{K}(G,X))_{K} over all sufficiently small compact subgroups is called a Shimura variety.


Surveys include

  • James Milne, Introduction to Shimura Varieties, 2004, (pdf)

  • Kai-Wen Lan, An Example-Based Introduction to Shimura Varieties, ([pdf])(

  • Wikipedia, Shimura variety

Introductory discussion of PEL Shimura varieties with an eye towards the definition of topological automorphic forms is in

Discussion in GAGA include

  • Kai-Wen Lan, Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties pdf

Discussion relating to moduli space of Calabi-Yau spaces includes

  • Alice Garbagnati, Bert van Geemen, Examples of Calabi-Yau threefolds parametrised by Shimura varieties (arXiv:1005.0478)

Model theoretic aspects

  • Boris Zilber, Model theory of special subvarieties and Schanuel-type conjectures, Annals of Pure and Applied Logic 167:10 (2016) 1000–1028 doi
  • C. Daw, A. Harris, Categoricity of modular and Shimura curves, J. Inst. Math. Jussieu, 16(5) (2017) 1075–1101 doi

Last revised on November 23, 2022 at 20:52:14. See the history of this page for a list of all contributions to it.