Abstract
Elsner, Luca and Tachiya proved in [4] that the values of the Jacobi-theta constants \(\theta_3(m\tau)\) and \(\theta_3(n\tau)\) are algebraically independent over \(\mathbb{Q}\) for distinct integers \(m\), \(n\) under some conditions on \(\tau\). On the other hand, in [3] Elsner and Tachiya also proved that three values \(\theta_3(m\tau),\theta_3(n\tau)\) and \(\theta_3(\ell \tau)\) are algebraically dependent over \(\mathbb{Q}\). In this article we prove the non-vanishing of linear forms in \(\theta_3(m\tau)\), \(\theta_3(n\tau)\) and \(\theta_3(\ell \tau)\) under various conditions on \(m\), \(n\), \(\ell\), and \(\tau\). Among other things we prove that for odd and distinct positive integers \(m,n>3\) the three numbers \(\theta_3(\tau)\), \(\theta_3(m\tau)\) and \(\theta_3(n \tau)\) are linearly independent over \(\overline{\mathbb{Q}}\) when \(\tau\) is an algebraic number of some degree greater or equal to 3. In some sense this fills the gap between the above-mentioned former results on theta constants. A theorem on the linear independence over \(\mathbb{C(\tau)}\) of the functions \(\theta_3(a_1 \tau), \dots, \theta_3(a_m \tau)\)for distinct positive rational numbers \(a_{1}, {\dots}, a_{m}\) is also established.
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Acknowledgements
We would like to express our deep gratitude to Professor Y.Tachiya for his useful comments. We are also indebted to the unknown reviewer for his helpful comments to improve the manuscript.
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Elsner, C., Kumar, V. On algebraic conditions for the non-vanishing of linear forms in Jacobi theta-constants. Acta Math. Hungar. 173, 392–413 (2024). https://doi.org/10.1007/s10474-024-01449-4
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DOI: https://doi.org/10.1007/s10474-024-01449-4