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j-line

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Mathematical concept Not to be confused with Line J (disambiguation).
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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()=\mathrm {Spec} (R)}

Here the j-invariant is normalized such that j = 0 {\displaystyle j=0} has complex multiplication by Z [ ζ 3 ] {\displaystyle \mathbb {Z} } , and j = 1728 {\displaystyle j=1728} has complex multiplication by Z [ i ] {\displaystyle \mathbb {Z} } .

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}} .

References

  1. Katz, Nicholas M.; Mazur, Barry (1985), Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, p. 228, ISBN 0-691-08349-5, MR 0772569.
  2. Gouvêa, Fernando Q. (2001), "Deformations of Galois representations", Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI, pp. 233–406, MR 1860043. See in particular p. 378.


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