TITLE: Hurwitz spaces and moduli spaces of marked elliptic curves
QUESTION [5 upvotes]: 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 ?

REPLY [4 votes]: 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.