Normal bases for modular function fields

TL;DR

This paper constructs a specific normal basis for a non-abelian Galois extension of modular function fields using Siegel functions, providing a concrete example in a complex setting.

Contribution

It presents the first explicit construction of a normal basis in a non-abelian modular function field extension using Siegel functions.

Findings

Constructed a normal basis for (X(N))/(X(1))

Used Siegel functions to explicitly find a free element

Provides a concrete example in non-abelian Galois extensions

Abstract

We provide a concrete example of a normal basis for a finite Galois extension which is not abelian. More precisely, let $\mathbb{C}(X(N))$ be the field of meromorphic functions on the modular curve $X(N)$ of level $N$. We construct a completely free element in the extension $\mathbb{C}(X(N))/\mathbb{C}(X(1))$ by means of Siegel functions.