# Explicit formula of a supersingular polynomial for rank-2 Drinfeld modules and applications

## TL;DR

This paper derives an explicit supersingular polynomial for rank-2 Drinfeld modules, highlighting their similarities to elliptic curves and applying this to construct optimal towers of Drinfeld modular curves.

## Contribution

It provides the first explicit formula for supersingular polynomials of rank-2 Drinfeld modules and explores their properties and applications.

## Key findings

- Explicit supersingular polynomial formula derived
- Basic properties of the polynomial proved
- Constructed an asymptotically optimal tower of Drinfeld modular curves

## Abstract

Rank-2 Drinfeld modules are a function-field analogue of elliptic curves, and the purpose of this paper is to investigate similarities and differences between rank-2 Drinfeld modules and elliptic curves in terms of supersingularity. Specifically, we provide an explicit formula of a supersingular polynomial for rank-2 Drinfeld modules and prove several basic properties. As an application, we give a numerical example of an asymptotically optimal tower of Drinfeld modular curves.