nLab Shimura variety

Idea

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

Definition

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

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

$$\Sh_{K}(G,X) \;\coloneqq\; G(\mathbb{Q}) \backslash G(\mathbb{A}_{f}) \times X/K$$

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 $\Sh(G,X):=(\Sh_{K}(G,X))_{K}$ 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 $\Sh(G,X)$ possesses an action of $G(\mathbb{A})$, as well as an action of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the theory of Shimura varieties finds application in the Langlands program. See also Youcis.

Related concepts

• Siegel modular form

• Hilbert modular form

• Rapoport-Zink space

References

General

Surveys:

• 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

• Tyler Lawson, An overview of abelian varieties in homotopy theory (arXiv:0810.0507)

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