Some Modular Varieties in Low Dimension Iii

Some years ago in [1] the first named Author and F. Hermann studied some modular varieties related to the orthogonal group O(2, n). The most significant variety they studied was related to O(2, 6), or —equivalently— to the symplectic group of degree two defined on the quaternions. Because of a mistake, they did not get a definite result regarding the structure of the graded ring of modular forms of degree two with respect to the Hurwitz integral quaternions. This has been obtained recently by Krieg in [4]. Then in [2], we reconsidered [1] and in its spirit, we got the structure of the graded ring of modular forms of degree two with respect to a congruence subgroup of the Hurwitz integral quaternions. As a consequence we got also Krieg’s results. Moreover in [1], it was explained a method for obtaining structure of the graded ring of modular forms related to O(2, n), n < 6, from the O(2, 6)-case. In the O(2, 5)case they illustrate substantially two cases. One has been solved in [2]. In this note we will approach and solve the second case. Our results generalze some results which have been obtained by Kl”ocker [Kl] in his doctoral thesis. This paper bases on many discussions with A. Krieg. We thank him for his support and for showing us unpublished manuscripts. We also thank I. Klöcker, who showed us an unpublished version of his thesis.