There are a number of different ways to define a Shimura variety.
One definition starts with connected complex manifolds such that for every point on the manifold there is an involution fixing only that point. If this manifold is isomorphic to a bounded open subset of $\C^n$ for some $n$, it is called a Hermitian symmetric domain (or space).
Then, (connected) Shimura varieties are quotients of Hermitian symmetric domains by torsion-free subgroups of the group of automorphisms of the domain.
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- Last edited by Bjorn Poonen on 2022-03-24 17:36:45