Hilbert quasimodular forms and vector bundles (Q515714)

In this paper, the author proves that the space of quasimodular Hilbert modular forms is isomorphic to the holomorphic sections of certain fiber bundles over the Hilbert modular variety, constructed explicitly. To do this, the author first identifies, by his previous work on quasimodular forms, the the space of quasimodular Hilbert modular forms with the similarly defined space of Hilbert quasimodular polynomials. Roughly these are polynomials whose coefficients are Hilbert quasimodular forms. Then he explicitly defines an action of \(\mathrm{SL}(2,\mathbb{R})^n\) on \(\mathbb{H}^n \times \mathbb{C}_\mu[X]\) by using the automorphy factors and related objects. The fiber bundle is then defined as \[ V_\mu := \Gamma \backslash (\mathbb{H}^n \times \mathbb{C}_\mu[X]), \] where \(\Gamma\) is the Hilbert modular group over a totally real field of degree \(n\).