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

1995/12/29 by Viacheslav V. Nikulin, Nikulin, Viacheslav V. · 1 citation
Mathematics · #10D20 #14D20 #17B65 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.alg-geom/9512018

openalex publication_date 1995/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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⊂ L_K3 on Picard lattice of K3'' for the corresponding moduli \M_S⊂ 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⊂ L_K3 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.

Citations

• A lecture on Kac--Moody Lie algebras of the arithmetic type

Cited by

• On the classification of hyperbolic root systems of the rank three. Part III

Related

• K3 surfaces with interesting groups of automorphisms
• Bloch's conjecture for (anti-)autoequivalences on K3 surfaces
• Automorphism groups in a family of K3 surfaces
• Mirror Symmetry, Borcherd-Harvey-Moore Products and Determinants of the Calabi-Yau Metrics on K3 Surfaces
• K3 surfaces with an automorphism of order 66, the maximum possible