2
$\begingroup$

For any $n\geq 1$, let $E_n $ be the elliptic curve given by the equation $y^2 = x(x-1)(x-\zeta_{15^n})$. Here $\zeta_{m} = \exp(2\pi i /m)$ for any positive integer $m$.

There is a unique element $\tau_n$ in the complex upper half plane such that $E_n = \mathbf{C}/(\mathbf{Z} + \tau_n\mathbf{Z})$.

I need that $\mathrm{Im}(\tau_n) > \frac{1}{2}$. Can we show this?

It might not be true, but if it is it would help me a lot.

  • 5
    $\tau_n$ is only unique up to the action of $SL(2,\mathbb{Z})$. If you choose it from the standard fundamental domain then $Im \tau_n\geq\sqrt{3}/2>1/2$.2011-10-13
  • 0
    If you have an ell. curve in Legendre form $y^2 = x(x-1)(x-\lambda)$ and $\lambda$ is close to $0$ (or $1$), why can we choose $\tau$ such that its imaginary part is greater than 1/2?2011-10-14

1 Answers 1