Quarter periodIn mathematics, the quarter periods K(m) and iK ′(m) are special functions that appear in the theory of elliptic functions. The quarter periods K and iK ′ are given by and When m is a real number, 0 < m < 1, then both K and K ′ are real numbers. By convention, K is called the real quarter period and iK ′ is called the imaginary quarter period. Any one of the numbers m, K, K ′, or K ′/K uniquely determines the others.
Abel elliptic functionsIn mathematics Abel elliptic functions are a special kind of elliptic functions, that were established by the Norwegian mathematician Niels Henrik Abel. He published his paper "Recherches sur les Fonctions elliptiques" in Crelle's Journal in 1827. It was the first work on elliptic functions that was actually published. Abel's work on elliptic functions also influenced Jacobi's studies of elliptic functions, whose 1829 published book "Fundamenta nova theoriae functionum ellipticarum" became the standard work on elliptic functions.
Modular groupIn mathematics, the modular group is the projective special linear group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane, which have the form where a, b, c, d are integers, and ad − bc = 1.
Elliptic curveIn mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K^2, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions (x, y) for: for some coefficients a and b in K. The curve is required to be non-singular, which means that the curve has no cusps or self-intersections.
