An algebraic characterization of the Kronecker function

The Kronecker function is a kind of generalization of the classical Eisenstein series which lies at the heart of Kronecker's theory of elliptic functions. Nowadays the Kronecker function is also known to play important roles in the geometry of configuration spaces of elliptic curves as well as periods of modular forms for PSL2(Z). In both of these contexts a key role is played by a certain three-term functional equation satisfied by the Kronecker function, the Fay identity. The goal of this talk is to explain the significance of this identity and how the Kronecker function is (up to mild ambiguity) uniquely determined by it.