Open Access

Weierstrass points on the Drinfeld modular curve \(\boldsymbol {X_{0}(\mathfrak {p})}\)

Research in the Mathematical Sciences20152:10

DOI: 10.1186/s40687-015-0026-9

Received: 30 September 2014

Accepted: 12 April 2015

Published: 10 July 2015

Abstract

Consider the Drinfeld modular curve \(X_{0}(\mathfrak {p})\) for 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 is a supersingular j-invariant. In this paper, we show the converse: Every supersingular j-invariant is the reduction modulo of the j-invariant of a Weierstrass point of \(X_{0}(\mathfrak {p})\).