Galois Theory and Modular Forms pp 3-12 | Cite as

# The Arithmetic of Weierstrass Points on Modular Curves *X*_{0} (*p*)

Chapter

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

## Keywords

Modular Form Cusp Form Compact Riemann Surface Weierstrass Point Modular Curf
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© Kluwer Academic Publishers 2004