modular discriminant

Definition 1.

Let ΛC be a lattice.

  1. 1.

    Let qτ=e2πiτ. The Dedekind eta functionMathworldPlanetmath is defined to be


    The Dedekind eta function should not be confused with the Weierstrass eta function, η(w;Λ).

  2. 2.

    The j-invariant, as a functionMathworldPlanetmath of lattices, is defined to be:


    where g2 and g3 are certain multiplesMathworldPlanetmath of the Eisenstein seriesMathworldPlanetmath of weight 4 and 6 (see entry).

  3. 3.

    The Δ function (delta function or modular discriminant) is defined to be


    Let Λτ be the lattice generated by 1,τ. The Δ function for Λτ has a product expansion

Title modular discriminant
Canonical name ModularDiscriminant
Date of creation 2013-03-22 13:54:09
Last modified on 2013-03-22 13:54:09
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Definition
Classification msc 33E05
Synonym delta function
Related topic EllipticFunction
Related topic JInvariant
Related topic WeierstrassSigmaFunction
Related topic DiscriminantMathworldPlanetmathPlanetmath
Related topic DiscriminantOfANumberField
Related topic RamanujanTauFunction
Defines modular discriminant
Defines Dedekind eta function