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Rational Values of Weierstrass Zeta Functions

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

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 ([email protected])
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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 

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