A Shimura variety is a higher-dimensional analog of a modular curve.
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
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.
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.
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
Discussion in GAGA include
Discussion relating to moduli space of Calabi-Yau spaces includes
Discussion related to the Langlands program can be found in
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.