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In Tom Apostols Proof of Picards Theorem [Modular Functions and Dirichlet Series in Number Theory, Thm. 2.10] he uses the Propertie of J' [Derivative of the J-Invariant] being only zero at $\rho$ and i and I get that those are in fact Zeros i just havent found an Argument that would suggest that those are the only ones.

Mathhead123
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    It depends on what you already know on the $j$ invariant, but it’s quite standard to see it’s a holomorphic map from $\mathbb{H}$ (the half-plane of complex numbers with positive imaginary part) onto $\mathbb{C}$, and the pre-image of a given $z$ is $SL_2(\mathbb{Z})\tau$ for some $\tau$ with positive imaginary part modulus at least one, and real part in $(-0.5,0.5]$. Now, any such $\tau$ not equal to $i$ or $e^{i\pi/3}$has an open neighborhood $U$ such that any $SL_2(\mathbb{Z})$ orbit meets $U$ at most once. Thus $j_{|U}$ is injective and this implies $j’(\tau) \neq 0$. – Aphelli Feb 16 '21 at 00:07
  • Ok got it, so the "Problem" with i and $\rho$ is that they include for every neighborhood $U$ at least two elements which are equivalent under $SL_2(\mathbb{Z})$ and thus j$|_U$ cannot be injective. – Mathhead123 Feb 16 '21 at 13:19
  • Yes, that’s exactly it! – Aphelli Feb 16 '21 at 15:17
  • See related https://math.stackexchange.com/q/3989856/72031 – Paramanand Singh Feb 18 '21 at 06:48

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