Transcendence of $πr$ or $\wp(ω_1 r)$

Yukitaka Abe

Published 2020-05-03Version 1

Let $\wp $ be a Weierstrass $\wp $-function with algebraic $g_2$ and $g_3$, whose fundamental periods $\omega _1, \omega _2$ satisfy ${\rm Im}(\omega _1) = 0$. We show that $\pi r$ or $\wp(\omega _1 r)$ is transcendental for any non-zero real number $r$.