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Picard–Fuchs Equation
In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves. Definition Let :j=\frac be the j-invariant with g_2 and g_3 the modular invariants of the elliptic curve in Weierstrass form: :y^2=4x^3-g_2x-g_3.\, Note that the ''j''-invariant is an isomorphism from the Riemann surface \mathbb/\Gamma to the Riemann sphere \mathbb\cup\; where \mathbb is the upper half-plane and \Gamma is the modular group. The Picard–Fuchs equation is then :\frac + \frac \frac + \frac y=0.\, Written in Q-form, one has :\frac + \frac f=0.\, Solutions This equation can be cast into the form of the hypergeometric differential equation. It has two linearly independent solutions, called the periods of elliptic functions. The ratio of the two periods is equal to the period ratio τ, the standard coordinate on the upper-half plane. However, the ratio of two solutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Forms
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. A modular function is a function that is invariant with respect to the modular group, but without the condition that be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are meromorphic (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function). Modular form theory is a special case of the more general theory of automorphic forms which are functions defined on Lie groups which transform nicely wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Functions
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass \wp-function. Further development of this theory led to hyperelliptic functions and modular forms. Definition A meromorphic function is called an elliptic function, if there are two \mathbb- linear independent complex numbers \omega_1,\omega_2\in\mathbb such that : f(z + \omega_1) = f(z) and f(z + \omega_2) = f(z), \quad \forall z\in\mathbb. So elliptic functions have two periods and are therefore also called ''doubly periodic''. Period lattice and fundamental domain Iff is an elliptic function with periods \omega_1,\omega_2 it also holds that : f(z+\gamma)=f(z) for every lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Manin Connection
In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space ''S'' of a family of algebraic varieties V_s. The fibers of the vector bundle are the de Rham cohomology groups H^k_(V_s) of the fibers V_s of the family. It was introduced by for curves ''S'' and by in higher dimensions. Flat sections of the bundle are described by differential equations; the best-known of these is the Picard–Fuchs equation, which arises when the family of varieties is taken to be the family of elliptic curves. In intuitive terms, when the family is locally trivial, cohomology classes can be moved from one fiber in the family to nearby fibers, providing the 'flat section' concept in purely topological terms. The existence of the connection is to be inferred from the flat sections. Intuition Consider a smooth morphism of schemes X\to B over characteristic 0. If we consider these spaces as complex analytic spaces, then the Ehresmann fibration theorem tell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
J-invariant
In mathematics, Felix Klein's -invariant or function, regarded as a function of a Complex analysis, complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is Holomorphic function, holomorphic away from a simple pole at the Cusp (singularity), cusp such that :j\left(e^\right) = 0, \quad j(i) = 1728 = 12^3. Rational functions of are modular, and in fact give all modular functions. Classically, the -invariant was studied as a parameterization of elliptic curves over , but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine). Definition The -invariant can be defined as a function on the upper half-plane :j(\tau) = 1728 \frac = 1728 \frac = 1728 \frac with the third definition implying j(\tau) can be expressed as a Cube (algebra), cube, also since 1728 (number), 1728 = 12^3. The given functions are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann P-function
In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and \infty. The equation is also known as the Papperitz equation. The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and \infty. That equation admits two linearly independent solutions; near a singularity z_s, the solutions take the form x^s f(x), where x = z-z_s is a local variable, and f is locally holomorphic with f(0)\neq0. The real number s is called the exponent of the solution at z_s. Let ''α'', ''β'' and ''γ'' be the exponents of one solution at 0, 1 and \infty respectively; and let ''α''', ''β''' and ''γ''' be those of the other. Then :\alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann's Differential Equation
In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and \infty. The equation is also known as the Papperitz equation. The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and \infty. That equation admits two linearly independent solutions; near a singularity z_s, the solutions take the form x^s f(x), where x = z-z_s is a local variable, and f is locally holomorphic with f(0)\neq0. The real number s is called the exponent of the solution at z_s. Let ''α'', ''β'' and ''γ'' be the exponents of one solution at 0, 1 and \infty respectively; and let ''α''', ''β''' and ''γ''' be those of the other. Then :\alpha + \alpha' + \beta + \beta' + \gamma + \gamma' = 1. By ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz Triangle Map
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necessarily a Schwarz triangle, although that is the most mathematically interesting case. When that triangle is a non-overlapping Schwarz triangle, i.e. a Möbius triangle, the inverse of the Schwarz triangle function is a single-valued automorphic function for that triangle's triangle group. More specifically, it is a modular function. Formula Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle in radians. Each of ''α'', ''β'', and ''γ'' may take values between 0 and 1 inclusive. Following Nehari, these angles are in clockwise order, with the vertex having angle ''πα'' at the origin and the vertex having angle ''πγ'' lying on the real line. The Schwarz triangle function can be given in terms of h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-period Ratio
In mathematics, the half-period ratio τ of an elliptic function is the ratio :\tau = \frac of the two half-periods \frac and \frac of the elliptic function, where the elliptic function is defined in such a way that :\Im(\tau) > 0 is in the upper half-plane. Quite often in the literature, ω1 and ω2 are defined to be the periods of an elliptic function rather than its half-periods. Regardless of the choice of notation, the ratio ω2/ω1 of periods is identical to the ratio (ω2/2)/(ω1/2) of half-periods. Hence the period ratio is the same as the "half-period ratio". Note that the half-period ratio can be thought of as a simple number, namely, one of the parameters to elliptic functions, or it can be thought of as a function itself, because the half periods can be given in terms of the elliptic modulus or in terms of the nome. See the pages on quarter period and elliptic integrals for additional definitions and relations on the arguments and parameters to ellipti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
