The arithmetic of Eisenstein series

In a first course on modular forms, one encounters the Eisenstein series $E_k$ of even weight $k>2$. More generally, Eisenstein series arise in real analytic families parametrized by a complex parameter s, and holomorphic Eisenstein series like $E_k$ are particular special values of $s$. 

The goal of this informal talk is to highlight the arithmetic information contained in some interesting examples of real analytic families of Eisenstein series at certain values of $s$. This is also an important theme of the 'Kudla program,' but I will try to focus on results that predate the current state-of-the-art research.