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On the values of Weierstrass zeta and sigma functions (with an appendix by David Masser)

Volume 208 / 2023

K. Senthil Kumar Acta Arithmetica 208 (2023), 285-294 MSC: Primary 11J81; Secondary 11J89. DOI: 10.4064/aa230201-22-5 Published online: 10 August 2023


We prove that if $\zeta (z)$ is a Weierstrass zeta function with algebraic invariants, then $\zeta (r)$ is transcendental for any positive rational number $r$ with at most one exception. This means that there are at most two nonzero rational numbers $r$ such that $\zeta (r)$ is algebraic: if $r$ is one of them, then the other one is $-r$. This comes tantalizingly close to resolving a problem mentioned by David Masser regarding the values of $\zeta (z)$ at rational numbers. We prove similar results for the Weierstrass sigma function. On the other hand, in an appendix, David Masser proves that there exist invariants $g_2,g_3$ such that $\zeta (3),\zeta (5)$ are rational. Note that in this case at least one of the resulting $g_2,g_3$ must be transcendental, by our result.


  • K. Senthil KumarNational Institute of Science Education and Research, Bhubaneswar
    An OCC of Homi Bhabha National Institute
    Khurda, 752050, Odisha, India

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