About: J-line

In the study of the arithmetic of elliptic curves, the j-line over a ring R is the coarse moduli scheme attached to the moduli problem sending a ring to the set of isomorphism classes of elliptic curves over . Since elliptic curves over the complex numbers are isomorphic (over an algebraic closure) if and only if their invariants agree, the affine space parameterizing j-invariants of elliptic curves yields a coarse moduli space. However, this fails to be a fine moduli space due to the presence of elliptic curves with automorphisms, necessitating the construction of the Moduli stack of elliptic curves.