TITLE: Is there a schemetical construction for modular curves over the rationals?
QUESTION [12 upvotes]: One can get modular curves by the following procedures: first take the uper half plane and the rationan numbers on the x-axis, then we consider the quotient by a congruence subgroup. Now we get a compact Riemann surface, and by Chow's results, it is algebraic--an algebraic variety. Here we start with an analytic object and finally we get an algebraic one. But can we use algebraic methords only (e.g. by quotients of group schemes) to get modular curves? Or, can we find a meanful moduli problem solved by a modular curve?

REPLY [3 votes]: Passing comment: Mumford's GIT constructs modular curves as quotients---not of the upper half plane, but of some parameter space of subspaces of projective space, by an algebraic group. As for the last part of your question: sure there is a moduli problem solved by a modular curve! Isomorphism classes of elliptic curves plus level structure (e.g. point of order N) is represented by a modular curve (Y_1(N) in this case, as long as N>=4).