On the values of Weierstrass zeta and sigma functions (with an appendix by David Masser)
Volume 208 / 2023
Abstract
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.
