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 {\displaystyle R} to the set of isomorphism classes of elliptic curves over R {\displaystyle R} . Since elliptic curves over the complex numbers are isomorphic (over an algebraic closure) if and only if their j {\displaystyle j} -invariants agree, the affine space A j 1 {\displaystyle \mathbb {A} _{j}^{1}} 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 Γ ( 1 ) {\displaystyle \Gamma (1)} in the following way:
M ( [ Γ ( 1 ) ] ) = S p e c ( R [ j ] ) {\displaystyle M([\Gamma (1)])=\mathrm {Spec} (R[j])}
Here the j-invariant is normalized such that j = 0 {\displaystyle j=0} has complex multiplication by Z [ ζ 3 ] {\displaystyle \mathbb {Z} [\zeta _{3}]} , and j = 1728 {\displaystyle j=1728} has complex multiplication by Z [ i ] {\displaystyle \mathbb {Z} [i]} . The j-line can be seen as giving a coordinatization of the classical modular curve of level 1, X 0 ( 1 ) {\displaystyle X_{0}(1)} , which is isomorphic to the complex projective line P / C 1 {\displaystyle \mathbb {P} _{/\mathbb {C} }^{1}} .
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