  Using the general method which was applied to prove finiteness of the set of
hyperbolic generalized Cartan matrices of elliptic and parabolic type, we
classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan
matrices of elliptic type and of rank 3 with a lattice Weyl vector.
  We develop the general theory of reflective lattices T with 2 negative
squares and reflective automorphic forms on homogeneous domains of type IV
defined by T. We consider this theory as mirror symmetric to the theory of
elliptic and parabolic hyperbolic reflection groups and corresponding
hyperbolic root systems. We formulate Arithmetic Mirror Symmetry Conjecture
relating both these theories and prove some statements to support this
Conjecture. This subject is connected with automorphic correction of Lorentzian
Kac--Moody algebras. We define Lie reflective automorphic forms which are the
most beautiful automorphic forms defining automorphic Lorentzian Kac--Moody
algebras, and we formulate finiteness Conjecture for these forms.
  Detailed study of automorphic correction and Lie reflective automorphic forms
for generalized Cartan matrices mentioned above will be given in Part II.
