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:
“Compare the history of the novel to that of rock ‘n’ roll. Both started out a minority taste, became a mass taste, and then splintered into several subgenres. Both have been the typical cultural expressions of classes and epochs. Both started out aggressively fighting for their share of attention, novels attacking the drama, the tract, and the poem, rock attacking jazz and pop and rolling over classical music.”
—W. T. Lhamon, U.S. educator, critic. “Material Differences,” Deliberate Speed: The Origins of a Cultural Style in the American 1950s, Smithsonian (1990)
“And out again I curve and flow
To join the brimming river,
For men may come and men may go,
But I go on forever.”
—Alfred Tennyson (1809–1892)