In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation
- Φn(x, y)=0,
where for the j-invariant j(τ),
- x=j(n τ), y=j(τ)
is a point on the curve. The curve is sometimes called X0(n), though often that is used for the abstract algebraic curve for which there exist various models. A related object is the classical modular polynomial, a polynomial in one variable defined as Φn(x, x).
It is important to note that the classical modular curves are part of the larger theory of modular curves. In particular it has another expression as a compactified quotient of the complex upper half-plane H.
Read more about Classical Modular Curve: Geometry of The Modular Curve, Parametrization of The Modular Curve, Mappings, Galois Theory of The Modular Curve
Famous quotes containing the words classical and/or curve:
“The basic difference between classical music and jazz is that in the former the music is always greater than its performance—Beethoven’s Violin Concerto, for instance, is always greater than its performance—whereas the way jazz is performed is always more important than what is being performed.”
—André Previn (b. 1929)
“In philosophical inquiry, the human spirit, imitating the movement of the stars, must follow a curve which brings it back to its point of departure. To conclude is to close a circle.”
—Charles Baudelaire (1821–1867)