Algebraic cycles and values of Green's functions

A conjecture of Gross, Kohnen and Zagier asserts that values of certain higher Green's functions are logarithms of algebraic numbers at CM points. This is now a theorem of Li.

In this talk we describe an alternative approach to this conjecture which relates it to the existence of higher Chow cycles in universal families over modular curves and show how to construct such cycles. The construction is related to classical 19th century work of Kummer and  Humbert as well as some ideas from enumerative geometry.