Generation of Siegel modular function fields of even level

Abstract

For positive integers g g and N N , let F N ( g ) \mathcal {F}_N^{(g)} be the field of meromorphic Siegel modular functions of genus g g and level N N whose Fourier coefficients belong to the N N th cyclotomic field. We construct explicit generators of F N ( g ) \mathcal {F}_N^{(g)} over F 1 ( g ) \mathcal {F}_1^{(g)} by making use of a quotient of theta constants, when g ≥ 2 g\geq 2 and N N is even.