View source for J-line

{{DISPLAYTITLE:''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 <math>R</math> to the set of isomorphism classes of elliptic curves over <math>R</math>. Since elliptic curves over the complex numbers are isomorphic (over an algebraic closure) if and only if their <math>j</math>-invariants agree, the affine space <math>\mathbb{A}^1_j</math> 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|congruence subgroup]] <math>\Gamma(1)</math> in the following way:<ref>{{citation
 | isbn = 0-691-08349-5
 | mr = 772569
 | page = 228
 | publisher = Princeton University Press, Princeton, NJ
 | series = Annals of Mathematics Studies
 | title = Arithmetic moduli of elliptic curves
 | url = https://books.google.com/books?id=M1IT0J_sPr8C&pg=PA228
 | volume = 108
 | year = 1985}}.</ref>

: <math>M([\Gamma(1)]) = \mathrm{Spec}(R[j]) </math>

Here the ''j''-invariant is normalized such that <math>j=0</math> has [[Complex multiplication|complex multiplication]] by <math>\mathbb{Z}[\zeta_3]</math>, and <math>j=1728</math> has complex multiplication by <math>\mathbb{Z}[i]</math>.

The ''j''-line can be seen as giving a coordinatization of the [[Classical modular curve|classical modular curve]] of level 1, <math>X_0(1)</math>, which is isomorphic to the [[Complex projective space|complex]] [[Projective line|projective line]] <math>\mathbb{P}^1_{/\mathbb{C}}</math>.<ref>{{citation
 | contribution = Deformations of Galois representations
 | mr = 1860043
 | pages = 233–406
 | publisher = Amer. Math. Soc., Providence, RI
 | series = IAS/Park City Math. Ser.
 | title = Arithmetic algebraic geometry (Park City, UT, 1999)
 | volume = 9
 | year = 2001}}. See in particular [https://books.google.com/books?id=PgHjLgIVidgC&pg=PA378 p.&nbsp;378].</ref>

==References==
{{reflist}}

[[Category:Elliptic curves]]



{{Sourceattribution|J-line}}