Projective Spaces as Orthogonal Modular Varieties

Abstract

We construct 16 reflection groups Γ acting on symmetric domains \(\mathcal {D}\) of Cartan type IV, for which the graded algebras of modular forms are freely generated by forms of the same weight, and in particular the Satake-Baily-Borel compactification of \(\mathcal {D}/ {\Gamma }\) is isomorphic to a projective space. Four of these are previously known results of Freitag-Salvati Manni, Matsumoto, Perna and Runge. In addition we find several new modular groups of orthogonal type whose algebras of modular forms are freely generated.