Weierstrass points on the Drinfeld modular curve $X_0(\mathfrak{p})$

TL;DR

This paper establishes a correspondence between supersingular $j$-invariants and Weierstrass points on Drinfeld modular curves, showing that every supersingular $j$-invariant arises from a Weierstrass point's reduction.

Contribution

It proves the converse of a known result, demonstrating that all supersingular $j$-invariants are reductions of Weierstrass points on $X_0(rak{p})$, a significant extension in the theory.

Findings

Every supersingular $j$-invariant is the reduction of a Weierstrass point's $j$-invariant.

The result links supersingular invariants with Weierstrass points on Drinfeld modular curves.

Enhances understanding of the structure of $X_0(rak{p})$ in characteristic $p$.

Abstract

Consider the Drinfeld modular curve $X_0(\mathfrak{p})$ for $\mathfrak{p}$ a prime ideal of $\mathbb{F}_q[T]$. It was previously known that if $j$ is the $j$-invariant of a Weierstrass point of $X_0(\mathfrak{p})$, then the reduction of $j$ modulo $\mathfrak{p}$ is a supersingular $j$-invariant. In this paper we show the converse: Every supersingular $j$-invariant is the reduction modulo $\mathfrak{p}$ of the $j$-invariant of a Weierstrass point of $X_0(\mathfrak{p})$.