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 $$R$$ to the set of isomorphism classes of elliptic curves over $$R$$. Since elliptic curves over the complex numbers are isomorphic (over an algebraic closure) if and only if their $$j$$-invariants agree, the affine space $$\mathbb{A}^1_j$$ 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.

This is related to the congruence subgroup $$\Gamma(1)$$ in the following way:


 * $$M([\Gamma(1)]) = \mathrm{Spec}(R[j]) $$

Here the j-invariant is normalized such that $$j=0$$ has complex multiplication by $$\mathbb{Z}[\zeta_3]$$, and $$j=1728$$ has complex multiplication by $$\mathbb{Z}[i]$$.

The j-line can be seen as giving a coordinatization of the classical modular curve of level 1, $$X_0(1)$$, which is isomorphic to the complex projective line $$\mathbb{P}^1_{/\mathbb{C}}$$.