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From Wikipedia, the free encyclopedia

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 to the set of isomorphism classes of elliptic curves over . Since elliptic curves over the complex numbers are isomorphic (over an algebraic closure) if and only if their -invariants agree, the affine space 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 in the following way:[1]

Here the j-invariant is normalized such that has complex multiplication by , and has complex multiplication by .

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

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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.


This page was last edited on 16 March 2023, at 21:19
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