Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T05:10:01.439Z Has data issue: false hasContentIssue false
  • English
  • Français

Rational Values of Weierstrass Zeta Functions

Part of: Model theory Diophantine approximation, transcendental number theory Other special functions

Published online by Cambridge University Press:  22 December 2015

G. O. Jones  and
M. E. M. Thomas
G. O. Jones
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
M. E. M. Thomas
Affiliation:
Zukunftskolleg, Fachbereich Mathematik und Statistik, Fach 216, Universitätsstraβe 10, Universität Konstanz, 78457 Konstanz, Germany (margaret.thomas@wolfson.oxon.org)
Get access Rights & Permissions [Opens in a new window]

Abstract

We answer a question of Masser by showing that for the Weierstrass zeta function ζ corresponding to a given lattice Λ, the density of algebraic points of absolute multiplicative height bounded by T and degree bounded by k lying on the graph of ζ, restricted to an appropriate domain, does not exceed c(log T)15 for an effective constant c > 0 depending on k and on Λ. Using different methods, we also give two bounds of the same form for the density of algebraic points of bounded height in a fixed number field lying on the graph of ζ restricted to an appropriate subset of (0, 1). In one case the constant c can be shown not to depend on the choice of lattice; in the other, the exponent can be improved to 12.

Keywords

Weierstrass zeta functionscountingirrationality

MSC classification

Secondary: 11J72: Irrationality; linear independence over a field 03C64: Model theory of ordered structures; o-minimality 33E05: Elliptic functions and integrals
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 4 , November 2016 , pp. 945 - 958
Copyright
Copyright © Edinburgh Mathematical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)