The transformation formula for a Siegel modular form can be interpreted as the statement that the modular form is a holomorphic section of a line bundle over the period domain (the quotient of the Siegel upper half-plane by a subgroup of finite index in the integral symplectic group). I've seen brief mentions of this fact in several references, but haven't been able to find one in which this geometric point of view is developped in detail. Does anyone know such a reference?


In particular, I would like to know if holomorphic line bundles over period domains have been classified. 


Thanks in advance



Siegel modular forms as sections of line bundles over the period domain
Modulo torsion, the Picard group of the quotient of the Siegel upper half plane by a finite-index subgroup of $\text{Sp}\_{2g}(\mathbb{Z})$ is just $\mathbb{Z}$. This result should probably be attributed to Borel. 


For a calculation of the torsion and explicit line bundles corresponding to the various pieces (plus references), see my paper "The Picard group of the moduli space of curves with level structures", available [here](http://www.nd.edu/~andyp/papers/). The results you want are discussed starting on page 3.


