J-line explained

J-line should not 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 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.\Gamma( 1 ) in the following way:^[1] M([\Gamma( 1 )]) = Spec (R[j])

Here the j-invariant is normalized such that

j =0 has complex multiplication by Z [\zeta _{3]} , and j =1728 has complex multiplication by 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 P ^{1} _{/ C} .^[2]

Notes and References

1. .
2. . See in particular p. 378.