A product formula for the higher rank Drinfeld discriminant function

TL;DR

This paper presents a generalized product formula for the higher rank Drinfeld discriminant function, improving computational efficiency and extending previous rank 2 results to arbitrary rank r.

Contribution

It introduces a new product expansion for the Drinfeld discriminant function in arbitrary rank, broadening the scope of previous formulas and enabling more efficient Fourier expansion calculations.

Findings

Provides a product expansion for arbitrary rank r

Enables more efficient Fourier expansion computations

Generalizes Gekeler's rank 2 formula to higher ranks

Abstract

We give a product expansion for the Drinfeld discriminant function in arbitrary rank $r$, which generalizes the formula obtained by Gekeler for the rank 2 Drinfeld discriminant function. This enables one to compute the Fourier expansion of this function much more efficiently. The formula in this article uses an $r-1$-dimensional parameter and as such provides a nice counterpoint to the formula previously obtained by Hamahata, which is written in terms of several 1-dimensional parameters.