Talk:Hilbert modular form

Unclear statements

The section Definition begins as follows:

"Let F be a totally real number field of degree m over the rational field. Let $\sigma _{1},\ldots ,\sigma _{m}$ be the real embeddings of F. Through them we have a map

$GL_{2}(F)\to GL_{2}(\mathbb {R} )^{m}.$

Let ${\mathcal {O}}_{F}$ be the ring of integers of F. The group $GL_{2}^{+}({\mathcal {O}}_{F})$ is called the full Hilbert modular group. For every element $z=(z_{1},\ldots ,z_{m})\in {\mathcal {H}}^{m}$, there is a group action of $GL_{2}^{+}({\mathcal {O}}_{F})$ defined by $\gamma \cdot z=(\sigma _{1}(\gamma )z_{1},\ldots ,\sigma _{m}(\gamma )z_{m})$"

Some problems:

1. The map $GL_{2}(F)\to GL_{2}(\mathbb {R} )^{m}$ is never explained clearly; there is never any good reason for omitting the important part of a definition,

2. The notation ${\mathcal {H}}$ is never defined. Is this supposed to mean the upper half plane, or in other words the hyperbolic plane? Then the article should use a word or two to say so.