The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras

Abstract

We find automorphic corrections for the Lorentzian Kac--Moody algebras with the simplest generalized Cartan matrices of rank 3: A_{1,0} = 2 0 -1 0 2 -2 -1 -2 2 and A_{1,I} = 2 -2 -1 -2 2 -1 -1 -1 2 For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form χ_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30 with respect to a 2-congruence subgroup. We find infinite product or sum expansions for these forms. Our method of construction of χ_{35} leads to the direct construction of Siegel modular forms by infinite product expansions, whose divisors are the Humbert surfaces with fixed discriminants. Existence of these forms was proved by van der Geer in 1982 using some geometrical consideration. We announce a list of all hyperbolic symmetric generalized Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.

40 pages, no figures. AMS-TeX

References in corpus (4)

• The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras
• Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras
• The Remark on Discriminants of K3 Surfaces Moduli as Sets of Zeros of Automorphic Forms
• Minimal Siegel modular threefolds

Cited by in corpus (37)

• A Farey tale for N=4 dyons
• Spectrum of Dyons and Black Holes in CHL orbifolds using Borcherds Lift
• The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras
• Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms
• K3 surfaces, modular forms, and non-geometric heterotic compactifications
• Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections
• A lecture on Arithmetic Mirror Symmetry and Calabi-Yau manifolds
• The Satake sextic in elliptic fibrations on K3
• Reflective modular forms in algebraic geometry
• Complex vector bundles and Jacobi forms
• Generalized Kac-Moody Algebras from CHL dyons
• Brewing moonshine for Mathieu
• 24 faces of the Borcherds modular form Phi_{12}
• BPS Saturated String Amplitudes: K3 Elliptic Genus and Igusa Cusp Form
• Automorphic Forms and Lorentzian Kac--Moody Algebras. Part I
• A Universal Genus-Two Curve from Siegel Modular Forms
• Arithmetic and Hyperbolic Structures in String Theory
• Borcherds Algebras and N=4 Topological Amplitudes
• Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds
• Second Quantized Mathieu Moonshine
• Enhanced Gauge Groups in N=4 Topological Amplitudes and Lorentzian Borcherds Algebras
• On the classification of hyperbolic root systems of the rank three. Part III
• Special Geometry and Automorphic Forms
• Jacobi Forms of Higher Index and Paramodular Groups in N=2, D=4 Compactifications of String Theory
• Inverse period mappings of $K3$ surfaces and a construction of modular forms for a lattice with the Kneser conditions
• Root systems for Lorentzian Kac-Moody algebras in rank 3 (arXiv version)
• Automorphic correction of the hyperbolic Kac-Moody algebra $E_{10}$
• Antisymmetric paramodular forms of weight 3
• A Theory of Lorentzian Kac-Moody algebras
• A Lecture about classification of Lorentzian Kac-Moody algebras of the rank three
• Weakly holomorphic modular forms and rank two hyperbolic Kac-Moody algebras
• Apollonian Packings and Kac-Moody Root Systems
• Two-Loop String Theory on Null Compactifications
• K3 surfaces with interesting groups of automorphisms
• A remark on algebraic surfaces with polyhedral Mori cone
• Borcherds Products Everywhere
• Mock Modularity Of Twisted Index In CHL Models