|
In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation :, such that is a point on the curve. Here denotes the . The curve is sometimes called , 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 . 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 . == Geometry of the modular curve == The classical modular curve, which we will call , is of degree greater than or equal to when , with equality if and only if is a prime. The polynomial has integer coefficients, and hence is defined over every field. However, the coefficients are sufficiently large that computational work with the curve can be difficult. As a polynomial in with coefficients in , it has degree , where is the Dedekind psi function. Since , is symmetrical around the line , and has singular points at the repeated roots of the classical modular polynomial, where it crosses itself in the complex plane. These are not the only singularities, and in particular when , there are two singularites at infinity, where and , which have only one branch and hence have a knot invariant which is a true knot, and not just a link. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「classical modular curve」の詳細全文を読む スポンサード リンク
|