The Arithmetic of Weierstrass Points on Modular Curves X ₀ (p)

Abstract

The purpose of this paper is to describe some recent results regarding the arithmetic properties of Weierstrass points on modular curves X ₀ (p) for primes p. We begin with some generalities; most of these can be found, for example, in the book of Farkas and Kra [F-K]. Suppose that X is a compact Riemann surface of genus g ≥2. If γ is a positive integer, then let H ^r (X) denote the space of holomorphic r-differentials on X. Each H ^r (X) is a finite-dimensional vector space over ℂ; we denote its dimension by d _r (X). A point Q ∈ X is called an r-Weierstrass point if there exists a non-zero differential w ∈ H ^r (X) such that

$${\operatorname{ord} _{Q\omega }} \geqslant {d_r}\left( X \right)$$