Siegel Modular Function

A Siegel modular function of degree is a quotient of scalar Siegel modular forms of the same weight, with not identically zero (Maass 1954-1955). These forms are holomorphic functions on the Siegel upper half-space , the space of symmetric complex matrices with positive definite imaginary part. For a block matrix in the integral symplectic group , their weight means

and the same transformation law holds for . In degree one the forms must also be holomorphic at the cusp; in higher degree this condition follows from their holomorphy and transformation law. The quotient is a meromorphic function on satisfying

For , this gives the usual modular functions, including meromorphy at the cusp. The Riemann theta function has transformation factors and is not itself such an invariant function.