# elliptic curve with Teichmüller parameter $τ=i$

Let the complex form $$z_{1}+dz_{1}\wedge dz_{2}$$ on $$D^{2}\times T^{2}$$ ,where $$D^2$$ is a unit disk un $$z_{1}$$-plane. here $$z_{1}$$ and $$z_{2}$$ is a standard coordinates on $$\mathbb{C^2}$$. I read on paper that on $$z_{1}=0$$ the complex structure is smooth elliptic curve with Teichmüller parameter $$τ=i$$.

Can you simple description to “smooth elliptic curve with Teichmüller parameter $$τ=i$$”?

I have introduction knowledge about complex geometry ie, if you describe it with standard complex coordinates, with standard 1-form or integrable complex structure , I understand very easily.

• Up to a constant there is only one holomorphic one-form on a complex elliptic curve, integrating it yields an isomorphism to $\Bbb{C/(u_1Z+u_2Z)}$ where $u_j = \int_{\gamma_j} \omega$ and $\gamma_1,\gamma_2$ generate the homology group, on the other hand, once your know $\gamma_1,\gamma_2$, to each $u_1,u_2$ there is a complex structure $S(u_1,u_2)$, and $S(u_1,u_2)$ is biholomorphic to $S(au_1+bu_2,cu_1+du_2),a,b,c,d\in \Bbb{Z}, ad-bc=1$. Here you have $(u_1,u_2) = (1,i)$. – reuns Jun 18 at 3:40
• It is not clear what does $z_1 + dz_1 \wedge dz_2$ mean. How to add a local function to a 2-form? – Arctic Char Jun 18 at 5:35
• @reuns first thank you very much for your answer, so two question:1. Why elliptic curve has only holomorphic bundle? 2. You say we have a space, where it is all elliptic curves up to biholomorphic, and this elliptic curve is $τ=i$ with SL(2,\mathbb{R}) natural action? – Ramtin.VA Jun 18 at 5:36
• @ArcticChar sorry I forgot to say $(z_{1},z_{2})$ is standard coordinates, i fix it now – Ramtin.VA Jun 18 at 5:38
• @reuns You are master and perfect on elliptic curves and complex geometry . Can you introduce to me what is best reference on elliptic curve and complex geometry? – Ramtin.VA Jun 18 at 5:43