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The question is from Washington's Elliptic Curves: Number Theory and Cryptography, Question 9.5.

The previous parts (which I have solved) are

(i) $\overline{j(\tau)}=j(-\overline{\tau})$,

(ii) if $\tau$ is in fundamental domain $\mathcal{F}$, then either $-\overline{\tau}\in\mathcal{F}$, or $\operatorname{re} \tau=-1/2$, or $|\tau|=1$ with $-1/2\leq \operatorname{re}(\tau)\leq 0$,

(iii) if $\tau\in\mathcal{F}$, $j(\tau)$ is real, then re $\tau=0$, re $\tau=-1/2$, or $|\tau|=1$ with $-1/2\leq \operatorname{re}(\tau)\leq 0$,

(iv) if $\tau\in\mathbb{H}$, $|\tau|=1$, then re $(-1/(\tau+1))=-1/2$,

(v) let $L$ be a lattice with $g_2(L)=-A$ and $g_3(L)=-B$, then there exists re $\tau'=0$ or $-1/2$, and $j(L)=j(\langle 1, \tau'\rangle)$.

The question I am stuck at is apparently the converse of (iv) in some sense. I have solved the case when re $\tau=0$, but am stuck when re $\tau=-1/2$. I am considering using some matrix in modular group $SL_2(\mathbb{Z})$, but the imaginary part of $\tau$ is giving me issues.

Thanks in advanced!

Kenny Wong
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    What is your definition of $g_2,g_3$ ? The statement is obvious both from the Eisenstein series definition (ie. $2$nd and $4$th derivative of $\wp_\tau(z)-1/z^2$) and the Fourier series with coefficients $\sigma_3(n),\sigma_5(n)$ – reuns Jan 18 '20 at 03:57
  • $g_2$ and $g_3$ are $60G_4$ and $280G_6$, where $G_4$ and $G_6$ are Eisenstein series. – Kenny Wong Jan 18 '20 at 05:53

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