Exceptional modular form of weight 4 on an exceptional domain contained in ${\mathbb{C}}^{2}7$

Abstract

Resnikoff [12] proved that weights of a non trivial singular modular form should be integral multiples of 1/2, 1, 2, 4 for the Siegel, Hermitian, quaternion and exceptional cases, respectively. The $\theta$-functions in the Siegel, Hermitian and quaternion cases provide examples of singular modular forms (Krieg [10]). Shimura [15] obtained a modular form of half-integral weight by analytically continuing an Eisenstein series. Bump and Baily suggested the possibility of applying an analogue of Shimura's method to obtain singular modular forms, i. e. modular forms of weight 4 and 8, on the exceptional domain of 3 x 3 hermitian matrices over Cayley numbers. The idea is to use Fourier expansion of a non-holomorphic Eisenstein series defined by using the factor of automorphy as in Karel [7]. The Fourier coefficients are the product of confluent hypergeometric functions as in Nagaoka [11] and certain singular series which we calculate by the method of Karel [6]. In this note we describe a modular form of weight 4 which may be viewed as an analogue of a $\theta$ zero-value and as an application, we consider its Mellin transform and prove a functional equation of the Eisenstein series which is a Nagaoka's conjecture (Nagaoka [11]).