On Siegel Modular Forms of Genus Two (II)

Abstract

Introduction. The main subject we shall discuss in this second paper is the Siegel modular forms of genus two with levels. The method we used in the first paper [5] did not give sufficient information even for level two. Therefore the problem (raised by Grothendieck) whether modular varieties become non-singular or not for higher levels was beyond our reach. With some other applications in mind, we therefore investigated as modular forms and proved, among others, a fundamental lemma in our recent paper [6]. Using the results in that paper, we shall show that modularvarieties of high levels do not have non-singular coverings even locally around their singular points. Also we shall determine how r2(1)/1'2(2) = Sp(2, Z/2Z) acts on the ring of modular forms A (r2 (2)) and obtain the characters of its action on the homogeneous parts A (r2 (2) ) k for k = 0, 1, 2, . In this way, we shall determine A (IF2 (1)) reproducing our earlier structure theorem on A (r2 (1 )) (2). Furthermore, the polynomial expressions of the four basic Eisenstein series of level one by theta-constants and a known identity of this kind (between a certain Eisenstein series of level two and the eighth power of Riemann's theta-constant [1]) will be obtained. We note that this identity was previously obtained using the Siegel main theorem on quadratic forms.