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

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

Here the j-invariant is normalized such that ${\displaystyle j=0}$ has complex multiplication by ${\displaystyle \mathbb {Z} [\zeta _{3}]}$, and ${\displaystyle j=1728}$ has complex multiplication by ${\displaystyle \mathbb {Z} [i]}$.

The j-line can be seen as giving a coordinatization of the classical modular curve of level 1, ${\displaystyle X_{0}(1)}$, which is isomorphic to the complex projective line ${\displaystyle \mathbb {P} _{/\mathbb {C} }^{1}}$.^[2]