IGUSA MODULAR-FORMS AND THE SIMPLEST LORENTZIAN KAC-MODY ALGEBRAS

Automorphic corrections for the Lorentzian Kac-Moody algebras with the simplest generalized Cartan matrices of rank 3, [GRAPHICS] are found. For A(1,0) this correction, which is a generalized Kac-Moody Lie supe ralgebra, is delivered by chi(35)(Z), the Igusa Sp(4)(Z)-modular form of weight 35, while for A(1,I) it is given by some Siegel modular form <(Delta)over tilde>(30)(Z) of weight 30 with respect to a 2-congruenc e subgroup of Sp(4)(Z). Expansions of chi(35)(Z) and <(Delta)over tild e>(30)(Z) in infinite products are obtained and the multiplicities of all the roots of the corresponding generalized Lorentzian Kac-Moody su peralgebras are calculated. These multiplicities are determined by the Fourier coefficients of certain Jacobi forms of weight 0 and index 1. The method adopted for constructing chi(35)(Z) and <(Delta)over tilde >(30)(Z) leads in a natural way to an explicit construction (as infini te products or sums) of Siegel modular forms whose divisors are Humber t surfaces with fi;red discriminants. A geometric construction of thes e forms was proposed by van der Geer in 1982. To show the prospects fo r further studies, the list of all hyperbolic symmetric generalized Ca rtan matrices A with the following; properties is presented: A is a ma trix of rank 3 and of elliptic; or parabolic type, has a lattice Weyl vector, and contains a parabolic submatrix (A) over tilde(1).