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

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

1995-12-29

arxiv.org/abs/alg-geom/9512018v3

Physics

High Energy Physics

High Energy Physics - Theory

AMSTeX, 6 pages, no figures. More exact formulations are given

Scientific paper

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.

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