Book contents
- Frontmatter
- Contents
- Preface
- 1 First Ideas: Complex Manifolds, Riemann Surfaces, and Projective Curves
- 2 Elliptic Integrals and Functions
- 3 Theta Functions
- 4 Modular Groups and Modular Functions
- 5 Ikosaeder and the Quintic
- 6 Imaginary Quadratic Number Fields
- 7 Arithmetic of Elliptic Curves
- References
- Index
4 - Modular Groups and Modular Functions
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 First Ideas: Complex Manifolds, Riemann Surfaces, and Projective Curves
- 2 Elliptic Integrals and Functions
- 3 Theta Functions
- 4 Modular Groups and Modular Functions
- 5 Ikosaeder and the Quintic
- 6 Imaginary Quadratic Number Fields
- 7 Arithmetic of Elliptic Curves
- References
- Index
Summary
The modular group of first level Γ1 = PSL(2, ℤ) made its debut in Section 2.6 in connection with the conformal equivalence of tori: Two such tori are equivalent if and only if their period ratios are related by a substitution of this kind. The quotient of the upper half-plane ℍ by the action of _1, and by the further modular groups to be introduced later in this section, produces a whole series of interrelated curves whose function fields (modular functions) have deep arithmetic and geometric applications going back to Abel [1827] and Jacobi [1829]. The two most striking of these are Hermite's solution of the general polynomial equation of degree 5 [1859], explained in Chapter 5, and Weber's realization [1891] of Kronecker's youthful dream (1860) of describing the absolute class field of the imaginary quadratic field for a positive square-free integer D; see Chapter 6. The present section prepares the way. Fricke and Klein [1926] and Shimura [1971] are recommended for information at a more advanced level.
The Modular Group of First Level
A modular function of first level is a function f of rational character in the open upper half-plane ℍ that is invariant under the action of Γ1 = PSL(2, ℤ), that is, f ((aω + b)/(cω + d)) = f(ω) for every ω ∈ ℍ and [ab/cd] ∈ Γ1; such a function is of period 1 because [11/01]: ω → ω + 1 is a modular substitution, so it is possible to expand it in powers of in the vicinity of, as in.
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- Elliptic CurvesFunction Theory, Geometry, Arithmetic, pp. 159 - 205Publisher: Cambridge University PressPrint publication year: 1997