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.
Surveys include
James Milne, Introduction to Shimura Varieties, 2004, (pdf)
Kai-Wen Lan, An Example-Based Introduction to Shimura Varieties, ([pdf])(https://www-users.cse.umn.edu/~kwlan/articles/intro-sh-ex.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
Last revised on November 23, 2022 at 20:52:14. See the history of this page for a list of all contributions to it.