I am an undergrad extremely interested in some applications of the Gross-Zagier formula for elliptic curves. I have a strong foundation in group theory and abstract algebra, and an understanding of elliptic curves at the level of *The Arithmetic of Elliptic Curves* by Silverman.

I get lost in the linked paper around the third paragraph ("There is a canonical construction of algebraic points...") when they start talking about imaginary quadratic fields, Hilbert class fields, and ideal class groups.

I want to know if there is a book, or set of books, that can teach me enough to understand the gross-zagier formula and all of the terms within it. I'm not expecting to understand it any time soon, but I'm not really sure where to look from here, or even what fields to explore. Perhaps Hartshorne?