I am trying to see that:
The moduli space of Riemann surfaces of genus 1 with one marked point is $\mathbb{H}/PSL(2,\mathbb{Z})$.
I know the facts that $PSL(2,\mathbb{Z})$ acts on the upper half plane $\mathbb{H}$ through holomorphic automorphisms and two elliptic curves $E_\tau, E_\mu$ are isomorphic $\tau$ and $\mu$ are related by a modular transformation (i.e. elements of $SL(2,\mathbb{Z}$).
It would be very helpful if some explains it from a completely analytic and geometric point of view. The definition of moduli space I am working with is: The moduli space $\mathfrak{M}_{g,n}$ is the set of isomorphism classes of Riemann surfaces of genus $g$ with $n$ reference points marked.
Best Answer
The upvoted comment above solves it for me. All credit goes to user10354138.