Generators of Siegel modular function field of higher genus and level

TL;DR

This paper explicitly constructs generators for the field of meromorphic Siegel modular functions of higher genus and level, using quotients of theta constants, expanding understanding of modular function fields.

Contribution

It provides explicit generators of the Siegel modular function field for higher genus and level using theta constants, a novel explicit description.

Findings

Explicit generators for $_N$ over $_1$ are given for $g extgreater 1$ and $N extgreater 2$.

Generators are expressed as quotients of theta constants.

The results extend the understanding of the structure of Siegel modular function fields.

Abstract

For positive integers $g$ and $N$, let $\mathcal{F}_N$ 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 present explicit generators of $\mathcal{F}_N$ over $\mathcal{F}_1$ in terms of quotient of theta constants, when $g\geq2$ and $N\geq 3$.