Timeline for Roots of modular functions

4 events

when toggle format	what		by	license	comment
Sep 16, 2018 at 18:22	vote	accept	Shimrod
Sep 16, 2018 at 11:24	answer	added	François Brunault		timeline score: 7
Sep 16, 2018 at 5:41	comment	added	reuns		$\Delta(\tau) = e^{2i \pi \tau} \prod_{n=1}^\infty (1-e^{2i \pi n \tau})^{24}$ has no zeros, it is real non-negative for $−i\tau >0$, and $\Delta(\tau+1)=\Delta(\tau), \Delta(−1/\tau)=\tau^{12}\Delta(\tau)=(−i\tau)^{12}\Delta(\tau)$. Thus, choosing the branch of $\Delta(\tau)^{1/d}$ and $(−i\tau)^{12/d}$ which are holomorphic for $\Im(\tau)>0$ and real non-negative for $−i\tau >0$, you'll have $\Delta(\tau+1)^{1/d}=e^{2i \pi / d}\Delta(\tau)^{1/d}$, $\Delta(−1/\tau)^{1/d}=(−i\tau)^{12/d}\Delta(\tau)^{1/d}$.
Sep 15, 2018 at 23:27	history	asked	Shimrod	CC BY-SA 4.0