Shimura variety

Summary

A Shimura variety is a quotient of a Hermitian symmetric space by a congruence subgroup of a reductive group over $\Q$. It is viewed as an algebraic variety. The family includes modular curves and the Siegel moduli spaces of abelian varieties. Shimura varieties are defined over number fields, and their cohomology carries both Galois actions and automorphic forms. This makes them a main bridge in the Langlands program.