mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Picard–Fuchs equation, named after

Émile Picard Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924. Life He was born in Paris on 24 July 1856 and educated there at th ...

and

Lazarus Fuchs Lazarus Immanuel Fuchs (5 May 1833 – 26 April 1902) was a Jewish-German mathematician who contributed important research in the field of linear differential equations. He was born in Mosina, Moschin in the Grand Duchy of Posen (modern-day M ...

, is a linear

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

whose solutions describe the periods of

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

Definition

Let :

j=\frac

be the

j-invariant In mathematics, Felix Klein's -invariant or function is a modular function of weight zero for the special linear group \operatorname(2,\Z) defined on the upper half-plane of complex numbers. It is the unique such function that is holomorphic a ...

with

g_2

and

g_3

the modular invariants of the elliptic curve in

Weierstrass form In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

: :

y^2=4x^3-g_2x-g_3.\,

Note that the ''j''-invariant is an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

from the

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

\mathbb/\Gamma

to the

Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...

\mathbb\cup\

; where

\mathbb

is the

upper half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...

and

\Gamma

is the

modular group In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...

. 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 In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...

. 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 solutions of the hypergeometric equation is also known as a Schwarz triangle map. The Picard–Fuchs equation can be cast into the form of Riemann's differential equation, and thus solutions can be directly read off in terms of Riemann P-functions. One has :

y(j)=P  \left\\,

At least four methods to find the j-function inverse can be given. Dedekind defines the ''j''-function by its Schwarz derivative in his letter to Borchardt. As a partial fraction, it reveals the geometry of the fundamental domain: :

2(S\tau)
(j) = \frac + \frac + \frac = \frac + \frac + \frac

where (''Sƒ'')(''x'') is the

Schwarzian derivative In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms an ...

of ''ƒ'' with respect to ''x''.

Generalization

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, this equation has been shown to be a very special case of a general phenomenon, the Gauss–Manin connection.

References

Pedagogical

* * J. Harnad and J. McKay, ''Modular solutions to equations of generalized Halphen type'', Proc. R. Soc. Lond. A 456 (2000), 261–294,

References

* J. Harnad, ''Picard–Fuchs Equations, Hauptmoduls and Integrable Systems'', Chapter 8 (Pgs. 137–152) of ''Integrability: The Seiberg–Witten and Witham Equation'' (Eds. H.W. Braden and I.M. Krichever, Gordon and Breach, Amsterdam (2000)). arXiv:solv-int/9902013 * For a detailed proof of the Picard-Fuchs equation: {{DEFAULTSORT:Picard-Fuchs Equation Elliptic functions Modular forms Hypergeometric functions Ordinary differential equations