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### Abstract

Most mathematicians have encountered modular functions. For example, when the
group theorists discovered the monster group, they were surprised to find that
the degrees of its irreducible representations were already encoded in the

*q*-coefficients of the

*j*-function. The theory of Shimura varieties grew out
of the applications of modular functions and modular forms to number theory.
Roughly speaking, Shimura varieties are the varieties on which modular
functions live.

### About

Notices of the AMS, December 2012.

### History

Posted May 13, 2012 (3 pages).

Minor revision September 1, 2012.