j-line

In the study of the arithmetic of elliptic curves, the j-line over any ring R is the coarse moduli scheme attached to the moduli problem Γ(1)]:^[1]

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

with the j-invariant normalized a la Tate: j = 0 has complex multiplication by Z[ζ₃], and j = 1728 by Z[i].

The j-line can be seen as giving a coordinatization of the classical modular curve of level 1, X₀(1), which is isomorphic to the complex projective line.^[2]

References

1. Katz, Nicholas M.; Mazur, Barry (1985), Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton University Press, Princeton, NJ, p. 228, ISBN 0-691-08349-5, MR 772569.
2. Gouvêa, Fernando Q. (2001), "Deformations of Galois representations", Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser. 9, Amer. Math. Soc., Providence, RI, pp. 233–406, MR 1860043. See in particular p. 378.