The Remark on Discriminants of K3 Surfaces Moduli as Sets of Zeros of Automorphic Forms

Abstract

We show that for any $N>0$ there exists a natural even $n>N$ such that the discriminant of moduli of K3 surfaces of the degree $n$ is not equal to the set of zeros of any automorphic form on the corresponding IV type domain. We give the necessary condition on a "condition $S\subset L_{K3}$ on Picard lattice of K3'' for the corresponding moduli $\M_{S\subset L_{K3}}$ of K3 to have the discriminant which is equal to the set of zeros of an automorphic form. We conjecture that the set of $S\subset L_{K3}$ satisfying this necessary condition is finite if $\rk S \le 17$. We consider this finiteness conjecture as "mirror symmetric'' to the known finiteness results for arithmetic reflection groups in hyperbolic spaces and as important for the theory of Lorentzian Kac--Moody algebras and the related theory of automorphic forms.