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 $${\operatorname{ord} _{Q\omega }} \geqslant {d_r}\left( X \right)$$ KeywordsModular FormCusp FormCompact Riemann SurfaceWeierstrass PointModular CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.