The distribution of Weierstrass points on a compact Riemann surface
type: Item
instance of: scholarly article
title: The distribution of Weierstrass points on a compact Riemann surface
zbMATH Open document ID: 0598.14016
DOI: 10.2307/2006944
author: Amnon Neeman
published in: Annals of Mathematics. Second Series
publication date: 1984-01-01T00:00:00Z
review text: The following beautiful result is proved: Let C be a compact Riemann surface of genus \(g\geq 2\) and let \(h:\quad C\to {\mathbb{R}}\) be a continuous, real valued function. Denote by \(J_{g-1+n}\) the jacobian of line bundles of degree \(g-1+n\) on C. Define a function \(Av^ nh\) on \(J_{g- 1+n}\) by the following formula: \[ (Av^ nh)(z)=\frac{1}{gn^ 2}\sum_{x\quad a\quad Weierstrass\quad point\quad of\quad z}h(x). \] Then as \(n\to \infty\), the \(Av^ nh\) converge uniformly to the constant function \(\int_{C}h db\) where b is the Bergman measure on C. The result strengthtens an unpublished result of D. Mumford which states that if L is a line bundle on C of positive degree, then the Cesaro limit \( Lim_{m\to \infty}(Av^*h)(L^ n)=\int_{C}h db.\) The author's proof is elegant and uses, in addition to the method of Mumford, several new and interesting techniques like studying \(Av^ nh\) via its Fourier coefficients and estimating high derivatives of the same function by pulling translation invariant vector fields back to the product of C with the theta divisor.
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 3964015
zbMATH Keywords: Weierstrass point
zbMATH Keywords: jacobian of line bundles
zbMATH Keywords: theta divisor
reviewed by: Dan Laksov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2006944
OpenAlex ID: W2317206451
description: scientific article; zbMATH DE number 3964015