Rank 2 symmetric hyperbolic Kac-Moody algebras and Hilbert modular forms

TL;DR

This paper explores rank two symmetric hyperbolic Kac-Moody algebras, associating them with Hilbert modular forms, and demonstrates automorphic corrections and root multiplicity formulas for specific cases.

Contribution

It establishes a connection between hyperbolic Kac-Moody algebras and Hilbert modular forms, providing automorphic corrections and asymptotic root multiplicity formulas.

Findings

Existence of chains of embeddings in families of H(a)

Automorphic correction for H(3), H(11), H(66) using Hilbert modular forms

Asymptotic formulas for root multiplicities based on Borcherds products

Abstract

In this paper we study rank two symmetric hyperbolic Kac-Moody algebras H(a) and their automorphic correction in terms of Hilbert modular forms. We associate a family of H(a)'s to the quadratic field Q(p) for each odd prime p and show that there exists a chain of embeddings in each family. When p = 5, 13, 17, we show that the first H(a) in each family, i.e. H(3), H(11), H(66), is contained in a generalized Kac-Moody superalgebra whose denominator function is a Hilbert modular form given by a Borcherds product. Hence, our results provide automorphic correction for those H(a)'s. We also compute asymptotic formulas for the root multiplicities of the generalized Kac-Moody superalgebras using the fact that the exponents in the Borcherds products are Fourier coefficients of weakly holomorphic modular forms of weight 0.