In mathematics, 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, is a linear

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...

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 ...

Definition

Let :

j=\frac

be the

j-invariant In mathematics, Felix Klein's -invariant or function, regarded as a function of a 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 holom ...

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 In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...

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 ve ...

\mathbb/\Gamma

to the

Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex number ...

\mathbb\cup\

; where

\mathbb

is the

upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...

and

\Gamma

is the

modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fraction ...

. 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 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-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, ...

s. 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 a ...

of ''ƒ'' with respect to ''x''.

Generalization

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

References

Pedagogical

* *

J. Harnad John Harnad (born ''Hernád János'') is a Hungary, Hungarian-born Canadian mathematical physicist. He did his undergraduate studies at McGill University and his doctorate at the University of Oxford (D.Phil. 1972) under the supervision of John ...

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