  Bruinier and Yang conjectured a formula for intersection numbers on an
arithmetic Hilbert modular surface, and as a consequence obtained a conjectural
formula for CM(K).G_1 under strong assumptions on the ramification in K. Yang
later proved this conjecture under slightly stronger assumptions on the
ramification. In recent work, Lauter and Viray proved a different formula for
CM(K).G_1 for primitive quartic CM fields with a mild assumption, using a
method of proof independent from that of Yang. In this paper we show that these
two formulas agree, for a class of primitive quartic CM fields which is
slightly larger than the intersection of the fields considered by Yang and
Lauter and Viray. Furthermore, the proof that these formulas agree does not
rely on the results of Yang or Lauter and Viray. As a consequence of our proof,
we conclude that the Bruinier-Yang formula holds for a slightly largely class
of quartic CM fields K than what was proved by Yang, since it agrees with the
Lauter-Viray formula, which is proved in those cases. The factorization of
these intersection numbers has applications to cryptography: precise formulas
for them allow one to compute the denominators of Igusa class polynomials,
which has important applications to the construction of genus 2 curves for use
in cryptography.
