Vol. 8, No. 1, 2019

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Algebraic results for the values $\vartheta_3(m\tau)$ and $\vartheta_3(n\tau)$ of the Jacobi theta-constant

Carsten Elsner, Florian Luca and Yohei Tachiya

Vol. 8 (2019), No. 1, 71–79
DOI: 10.2140/moscow.2019.8.71
Abstract

Let ϑ3(τ) = 1 + 2 ν=1eπiν2τ denote the classical Jacobi theta-constant. We prove that the two values ϑ3(mτ) and ϑ3(nτ) are algebraically independent over for any τ in the upper half-plane such that q = eπiτ is an algebraic number, where m,n 2 are distinct integers.

Keywords
algebraic independence, Jacobi theta-constants, modular functions
Mathematical Subject Classification 2010
Primary: 11J85
Secondary: 11J91, 11F27
Milestones
Received: 10 January 2018
Revised: 18 June 2018
Accepted: 3 July 2018
Published: 11 August 2018
Authors
Carsten Elsner
Fachhochschule für die Wirtschaft
University of Applied Sciences
Hannover
Germany
Florian Luca
School of Mathematics
University of the Witwatersrand
Johannesburg
South Africa
Max Planck Mathematical Institute
Bonn
Germany
Department of Mathematics
Faculty of Sciences
University of Ostrava
Ostrava
Czech Republic
Yohei Tachiya
Graduate School of Science and Technology
Hirosaki University
Hirosaki
Japan