#### 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 ${\vartheta }_{3}\left(\tau \right)=1+2{\sum }_{\nu =1}^{\infty }{e}^{\pi i{\nu }^{2}\tau }$ denote the classical Jacobi theta-constant. We prove that the two values ${\vartheta }_{3}\left(m\tau \right)$ and ${\vartheta }_{3}\left(n\tau \right)$ are algebraically independent over $ℚ$ for any $\tau$ in the upper half-plane such that $q={e}^{\pi i\tau }$ is an algebraic number, where $m,n\ge 2$ are distinct integers.

##### Keywords
algebraic independence, Jacobi theta-constants, modular functions
##### Mathematical Subject Classification 2010
Primary: 11J85
Secondary: 11J91, 11F27