Generation of Siegel modular function fields of even level

Abstract:

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