I was wondering if anything is known about the roots of $J' = \frac{dJ}{d\tau}$. Here, $J(\tau) = j(\tau)/12^3$ is Klein's absolute invariant. Some roots can be calculated (at least I know how to), but is anything general known?
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Using Ramanujan's system of ODEs, you can express $\frac{1}{2\pi\mathrm{i}}\frac{\mathrm{d}j}{\mathrm{d}\tau}$ as a rational function of Eisenstein series; the denominator is the modular discriminant, and the numerator holds the answer to your question. – ccorn Aug 09 '16 at 19:28
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The solution here is much easier than the one to that question. – ccorn Aug 09 '16 at 19:31
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@ccorn Using Ramanujan's differential equations we get $J'/J = -2\pi i E_6(\tau)/E_4(\tau)$. Are the roots of $E_6(\tau)$ known? Do they all have explicit form or are they all in a certain region? Is anything general known? – glebovg Aug 12 '16 at 01:48
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@ccorn I came across a theorem of Rankin and Swinnerton-Dyer, which was interesting. – glebovg Aug 12 '16 at 16:38
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Correct so far; multiply with the expression of $J$ in terms of Eisenstein series to get the full numerator relevant for $J'$. As for the zeros of $E_4$ and $E_6$, this has been covered (indirectly) e.g. here and there. I suppose you can then present the answer yourself, I am looking forward to that. – ccorn Aug 12 '16 at 16:38