nLab Shimura variety

Redirected from "Shimura varieties".
Contents

Contents

Idea

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

Definition

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) \;\coloneqq\; 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.

Applications to the Langlands program

Since the l-adic cohomology of the Shimura variety Sh(G,X)\Sh(G,X) possesses an action of G(𝔸)G(\mathbb{A}), as well as an action of the absolute Galois group Gal(¯/)\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}), the theory of Shimura varieties finds application in the Langlands program. See also Youcis.

References

General

Surveys:

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)

Discussion related to the Langlands program can be found in

  • Alex Youcis, The cohomology of Shimura varieties and the Langlands correspondence (pdf)

Model theoretic aspects

Discussion in model theory:

  • 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 July 23, 2023 at 07:14:52. See the history of this page for a list of all contributions to it.