Mathematics – Number Theory
Scientific paper
2006-04-29
Mathematics
Number Theory
30 pages. Article gives summary of main results of the e-print math.NT/0605012, with addition of background information and mo
Scientific paper
Let $G=SO(3,C)$, $\Gamma=SO(3,Z[i])$, $K=SO(3)$, and let $X$ be the locally symmetric space $\Gamma\backslash G/K$. In this paper, we write down explicit equations defining a fundamental domain for the action of $\Gamma$ on $G/K$. The fundamental domain is well-adapted for studying the theory of $\Gamma$-invariant functions on $G/K$. We write down equations defining a fundamental domain for the subgroup $\Gamma_Z=SO(2,1)_Z$ of $\Gamma$ acting on the symmetric space $G_{R}/K_R$, where $G_R$ is the split real form SO(2,1) of $G$ and $K_R$ is its maximal compact subgroup SO(2). We formulate a simple geometric relation between the fundamental domain of $\Gamma$ and $\Gamma_Z$ so described. These fundamental domains are geared towards the detailed study of the spectral theory of $X$ and the embedded subspace $X_R=\Gamma_Z\backslash G_R/K_R$.}
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