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

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 ⊂ LK3 on Picard lattice of K3” for the corresponding moduli MS⊂LK3 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 ⊂ LK3 satisfying this necessary condition is finite if rk S ≤ 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. 0. Introduction. The subject of this paper is directly connected with the recently developed theory of Lorentzian Kac–Moody algebras, the corresponding theory of automorphic forms on IV type domains, related geometry of K3 surfaces and their moduli, and mirror symmetry. See [Bo1], [Bo2], [Bo3], [G1],[G2], [G3], [G4], [GN1], [GN2], [GN3], [N9], [N10]. We refer a reader who is interested in physical applications to the recent papers [H–M] and [Ka]. Results of this article were inspired by my thinking over recent announcements by J. Jorgensen and A. Todorov on discriminants of K3 surfaces. I am grateful to Prof. V.A. Gritsenko for very useful discussions on this subject. 1. Definitions. Let LK3 be an even hyperbolic unimodular lattice of the signature (3, 19). It is known that LK3 is isomorphic to the H (X ;Z) for a K3 surface X over C. To be shorter, we set L =: LK3 below. Let S ⊂ L be a primitive hyperbolic sublattice, i.e. S is a lattice of the signature (1, k) and L/S is a free Z-module. The pair S ⊂ L is called ” a condition on Picard lattice of K3” (or, shortly ”condition S ⊂ L”). We correspond to the condition S ⊂ L a lattice T (S ⊂ L) = S L of the signature (2, 20 − rk S) and a complex symmetric domain of the type IV Ω(S ⊂ L) = {Cω ⊂ T (S ⊂ L)⊗ C | ω = 0, ω · ω > 0} (1.1) of the dimension 20− rk S. Any e ∈ S ⊗ R with e < 0 defines a codimension one symmetric subdomain of the type IV De = {Cω ∈ Ω(S ⊂ L) | ω · e = 0}. (1.2) Partially supported by Grant of Russian Fund of Fundamental Research. 2 VIACHESLAV V. NIKULIN The subset D̃(S ⊂ L) = ⋃ δ∈T (S⊂L), δ=−2 Dδ (1.3) is called the discriminant of the condition S ⊂ L. We will consider automorphic forms (i.e., holomorphic, with some non-negative weight) on Ω(S ⊂ L) with respect to subgroups of finite index G ⊂ O(T (S ⊂ L)). (1.4) If there exists such an automorphic form Φ of positive weight with the set of zeros D̃(S ⊂ L), we say that the discriminant D̃(S ⊂ L) is equal to the set of zeros of an automorphic form. We remark that for this definition, we don’t fix the subgroup G of finite index. We want to show that this situation is extremely rare if rk S ≤ 17. The situation which is described above is purely arithmetic and one can forget about S ⊂ L considering only the lattice T = T (S ⊂ L). Considering of S ⊂ L is important for the theory of K3 surfaces. One can correspond to the condition on Picard lattice of K3 S ⊂ L a family π : X S⊂L → MS⊂L of K3 surfaces where MS⊂L = G \ Ω(S ⊂ L). For this family, Xm = π (m) is a non-singular K3 surface if m ∈ MS⊂L = G \ (Ω(S ⊂ L) − D̃(S ⊂ L)), and π (m) is a singular K3 surface with Du Val singularities if m ∈ DS⊂L = G \ D̃(S ⊂ L). For a general point m ∈ MS⊂L, the embedding SXm ⊂ H (Xm;Z) of Picard lattice SXm of Xm is isomorphic to S ⊂ L. This K3 interpretation follows (see [N1], [N2]) from general results [Tjurina, S̆], [P-S̆–S̆], [B–R], [Ku]. Automorphic forms on Ω(S ⊂ L) give sections of appropriate ”linear” sheafs on MS⊂L. Thus, geometrically, the discriminant D̃(S ⊂ L) is the set of zeros of an automorphic form if DS⊂L is the set of zeros of a section of an appropriate ”linear” sheaf on MS⊂L. We don’t specify the group G since there are several natural choices of this group. If the discriminant D̃(S ⊂ L) is equal to the set of zeros of an automorphic form, then for any subgroup G ⊂ O(T (S ⊂ L)) of finite index the preimage of the discriminant DS⊂L is the set of zeros of a section of an appropriate ”linear” sheaf on some finite ramified covering of MS⊂L.