J-line

In the study of the arithmetic of elliptic curves, the ''j''-line over a ring (mathematics), 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 $\mathbb^1_j$ parameterizing j-invariant, 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:

: $M([\Gamma(1)]) = \mathrm(R[j])$

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

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 space, complex projective line $\mathbb^1_$.. See in particula
p. 378

References

Moduli theory
Elliptic curves

{{math-stub