Shimura Varieties

Shimura varieties are mathematical objects that arise in the fields of algebraic geometry and number theory. They serve as spaces that help mathematicians study complex structures, particularly those related to symmetries and group actions. Essentially, they generalize certain types of geometric shapes and are connected to number theory through their relationships with automorphic forms, which are functions that exhibit specific symmetry properties. These varieties play a crucial role in modern theories, linking areas such as arithmetic geometry, representation theory, and the Langlands program, which seeks deep connections between number theory and harmonic analysis.