j-line

In the study of the arithmetic of elliptic curves, the j-line over any ring R is the coarse moduli scheme attached to the moduli problem Γ(1)]:^[1]

M([\Gamma (1)])=\mathrm {Spec} (R[j])

with the j-invariant normalized a la Tate: j = 0 has complex multiplication by Z[ζ₃], and j = 1728 by Z[i].

The j-line can be seen as giving a coordinatization of the classical modular curve of level 1, X₀(1), which is isomorphic to the complex projective line.^[2]

References

^ 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.
^ 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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[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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