Question

I would like to know if the moduli space $\mathcal M_{1,n}$ of genus $1$ curves with $n$ marked points can be realized as a Hurwitz space ?

Answer 1

This might depend on what you count as a "Hurwitz space." To me, any cover of M_{g,n} parametrizing covers branched at the marked points is a Hurwitz space; so I would say, in a tautological tone of voice, that M_{1,n} is a Hurwitz space parametrizing degree-1 covers of elliptic curves branched at the n marked points!

But maybe you really want M_{1,n} to be a moduli space of branched covers of P^1? This seems plausible. It might be a pain to do in practice. I suppose I would try to set it up as a moduli space of covers Y -> P^1 which factor as Y -> E -> P^1, and where Y -> E is branched at the n marked points. But then you'll have to worry about collisions between marked points and Weierstrass points... it sounds like a pain.