  We find automorphic corrections for the Lorentzian Kac--Moody algebras with
the simplest generalized Cartan matrices of rank 3:
  A_{1,0} =
  2 0 -1
  0 2 -2
  -1 -2 2
  and
  A_{1,I} =
  2 -2 -1
  -2 2 -1
  -1 -1 2
  For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form
\chi_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30
with respect to a 2-congruence subgroup. We find infinite product or sum
expansions for these forms. Our method of construction of \chi_{35} leads to
the direct construction of Siegel modular forms by infinite product expansions,
whose divisors are the Humbert surfaces with fixed discriminants. Existence of
these forms was proved by van der Geer in 1982 using some geometrical
consideration. We announce a list of all hyperbolic symmetric generalized
Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a
lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.

Algebraic Geometry;High Energy Physics - Theory;Algebraic Geometry;Quantum Algebra;Quantum Algebra