j-line
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 [math]\displaystyle{ R }[/math] to the set of isomorphism classes of elliptic curves over [math]\displaystyle{ R }[/math]. Since elliptic curves over the complex numbers are isomorphic (over an algebraic closure) if and only if their [math]\displaystyle{ j }[/math]-invariants agree, the affine space [math]\displaystyle{ \mathbb{A}^1_j }[/math] 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 [math]\displaystyle{ \Gamma(1) }[/math] in the following way:[1]
- [math]\displaystyle{ M([\Gamma(1)]) = \mathrm{Spec}(R[j]) }[/math]
Here the j-invariant is normalized such that [math]\displaystyle{ j=0 }[/math] has complex multiplication by [math]\displaystyle{ \mathbb{Z}[\zeta_3] }[/math], and [math]\displaystyle{ j=1728 }[/math] has complex multiplication by [math]\displaystyle{ \mathbb{Z}[i] }[/math].
The j-line can be seen as giving a coordinatization of the classical modular curve of level 1, [math]\displaystyle{ X_0(1) }[/math], which is isomorphic to the complex projective line [math]\displaystyle{ \mathbb{P}^1_{/\mathbb{C}} }[/math].[2]
References
- ↑ Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, 108, Princeton University Press, Princeton, NJ, 1985, p. 228, ISBN 0-691-08349-5, https://books.google.com/books?id=M1IT0J_sPr8C&pg=PA228.
- ↑ "Deformations of Galois representations", Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., 9, Amer. Math. Soc., Providence, RI, 2001, pp. 233–406. See in particular p. 378.
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