# Interpretation of the Gross-Zagier formula for Green function

I am reading the paper of Gross and Zagier on heights of Heegner points and would like to check with the experts whether the following (meta?)mathematical statement makes sense.

In the calculation of the Green function on the modular curve $X_0(N)$ Gross and Zagier work with the open part of the curve and then take care of the cusps. Can this be interpreted as doing Arakelov geometry on the "blowup of the compactification of the modular curve over $\rm Spec({\mathbb Z})$ at the points that are cusps over the archimedian place". It is reasonable that there is a projection formula so that the heights (squares) of the divisor on different compactifications and easily compared.

Still, I am not certain where the parameter $s$ comes from. Perhaps, blowups of this kind, similar to blowups in symplectic geometry come with a parameter?