This monograph treats one case of a series of conjectures by S. Kudla whose goal is to show
that Fourier of Eisenstein series encode information about the Arakelov intersection theory of
special cycles on Shimura varieties of orthogonal and unitary type. Here the Eisenstein series
is a Hilbert modular form of weight one over a real quadratic field the Shimura variety is a
classical Hilbert modular surface and the special cycles are complex multiplication points and
the Hirzebruch-Zagier divisors. By developing new techniques in deformation theory the authors
successfully compute the Arakelov intersection multiplicities of these divisors and show that
they agree with the Fourier coefficients of derivatives of Eisenstein series.
