AUTOMORPHIC FORMS ON THE COMPLEX AND REAL BALLS DERIVED FROM THETA CONSTANTS

Abstract

Through the modular embedding of the complex n-dimensional ball BnC into the Siegel upper half-space Sn+1 of degree n + 1 with respect to the Eisenstein integers Z[ω], we pull back the theta constants on Sn+1. We find a condition on the characteristics of the theta constants so that the pullbacks are non-zero automorphic forms on BnC with respect to the congruence subgroup Γ(1−ω). These automorphic forms are real valued on the real ball naturally embedded in the complex ball.