Multiplier systems for Hermitian modular groups

TL;DR

This paper investigates the existence and properties of multiplier systems for Hermitian modular groups, especially focusing on half-integer weights and their kernels, revealing connections to non-congruence subgroups and explicit examples.

Contribution

It establishes the existence conditions for multiplier systems of weight 1/r, describes the kernel of associated characters, and links to explicit constructions like Wang's Borcherds product.

Findings

Multiplier systems of weight 1/r exist only for r=1 or 2.

The kernel of the character on the unimodular group is a non-congruence subgroup.

For small groups, the kernel matches known non-congruence groups by Kubota and others.

Abstract

Let $\Gamma_{F,n}$ be the Hermitian modular group of degree $n>1$ in sense of Hel Braun with respect to an imaginary quadratic field $F$. Let $r$ be a natural number. There exists a multiplier system of weight $1/r$ (equivalently a Hermitian modular form of weight $k+1/r$, $k$ integral) on some congruence group if and only if $r=1$ or $2$. This follows from a much more general construction of Deligne [De] combining it with results of Hill [Hi], Prasad [P] and Prasad-Rapinchuk [PR]. As far as we know, the systems of weight $1/2$ have not yet been described explicitly. Remarkably Haowu Wang [Wa] gave an example of a modular form of half integral weight. Actually he constructs a Borcherds product of weight $23/2$ for a group of type $O(2,4)$. This group is isogenous to the group $U(2,2)$ that contains the Hermitian modular groups of degree two. In this paper we want to study such multiplier systems. If one restricts them to the unimodular group one obtains a usual character. Our main result states that the kernel of this character is a non-congruence subgroup. For sufficiently small $\Gamma$ it coincides with the group described by Kubota in the case $n=2$ and by Bass Milnor Serre in the case $n>2$.