j-line

Not to be confused with Line J (disambiguation).

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 $\mathbb {A} _{j}^{1}$ 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 $\mathbb {P} _{/\mathbb {C} }^{1}$.