What is the Weil-Petersson metric of the moduli space of elliptic curves?

Question

One can define the Weil-Petersson metric on the moduli space of Riemann surfaces. I would like to know an explicit example of such a metric. What is the Weil-Petersson metric of the moduli space of elliptic curves (which can be thought of a punctured sphere or the upper half plane)?

Answer 1

So, if I understand you correctly, let $S = T^2$ be the real torus of dimension 2. You are wondering what the Weil-Petersson metric is on the moduli space of complex structures ${\cal M}(S)$.

The Teichmüller space of $S$ denoted ${\cal T}(S)$ can naturally be identified with the upper half-plane $\mathbb{H}^2$. Under this identification, the Weil-Petersson metric on ${\cal T}(S)$ is just the hyperbolic metric on $\mathbb{H}^2$. I have to confess, I don't know what is the simplest way to see that this is true.

Then the moduli space ${\cal M}(S)$ is the modular curve $\mathbb{H}^2 / SL(2,\mathbb{Z})$, an the Weil-Petersson metric there is just the induced hyperblic metric on the quotient.

I hope I did not say anything wrong (and answered your question), some people would be far more competent than me to answer this.