The Elliptic Modular Functions Associated With the Elliptic Norm Curve E7 (Classic Reprint)

Explore the mathematics of elliptic modular functions through a focused study of the elliptic norm curve E7.

This work offers a rigorous look at how the modular group acts on curves, the structure of their symmetries, and the resulting elliptic functions that describe their geometry. Suitable for readers comfortable with higher-level algebra and geometry, it illuminates the n = 7 case as a representative, more complex example of the general theory.

The author develops the group-theoretic and geometric framework step by step, deriving transformations, invariants, and the fundamental forms that encode the family of curves. Key ideas include the Kleinian form, a fixed-space viewpoint, and the connections between modular functions and the coordinates that parametrize the curves. The text also discusses rational curves, loci of special points, and the role of Klein’s quartic in understanding these modular objects.

• A clear construction of the symmetry groups acting on the elliptic norm curve and their implications for birational maps.
• Derivations of the fundamental elliptic modular functions and their algebraic properties.
• The Kleinian framework, including the Klein quartic and its relation to the modular locus and fixed spaces.
• Parametric representations and covariant forms that link geometric structures to modular functions.

Ideal for readers with a background in algebraic geometry, modular forms, or geometric group theory who want a detailed, methodical treatment of elliptic modular functions tied to E7.