  Around 2000 Kudla presented conjectures about deep relations between
arithmetic intersection theory, Eisenstein series and their derivatives, and
special values of Rankin $L-$series. The aim of this text is to work out the
details of an old unpublished draft on the second author's attempt to prove
these conjectures for the case of the product of two modular curves.
  In part one we proved that the generating series of certain modified
arithmetic special cycles is as predicted by Kudla's conjectures a modular form
with values in the first arithmetic Chow group. Here we pair this generating
series with the square of the first arithmetic Chern class of the line bundle
of modular forms. Up to previously known Faltings heights of Hecke
correspondences only integrals of the Green functions $\Xi (m)$ over $X$ had to
be computed. The resulting arithmetic intersection numbers turn out to be as
predicted by Kudla to be strongly related to the Fourier coefficients of the
derivative of the classical real analytic Eisenstein series $E_2(\tau, s)$.
