Picard–Fuchs equation: Difference between revisions

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{{Short description|Mathematical equation}}In [[mathematics]], the '''Picard-FuchsPicard–Fuchs equation''', named after [[Émile Picard]] and [[Lazarus Fuchs]], is a linear [[ordinary differential equation]] whose solutions describe the periods of [[elliptic curve]]s.

==Definition==

Let

:<math>j=\frac{g_2^3}{g_2^3-27g_3^2}</math>

be the [[j-invariant]] with <math>g_2</math> and <math>g_3</math> the [[Elliptic modular function|modular invariantinvariants]]s of the elliptic curve in [[Weierstrass equation|Weierstrass form]]:

:<math>y^2=4x^3-g_2x-g_3.\,</math>

Note that the ''j''-invariant is an [[isomorphism]] from the [[Riemann spheresurface]] <math> \mathbb{CH}/\cup\{\infty\}Gamma </math> to the [[Riemann surfacesphere]] '''H'''<math>\mathbb{C}\cup\{\infty\}</&Gammamath>; where '''<math>\mathbb{H'''}</math> is the [[upper half-plane]] and &<math>\Gamma;</math> is the [[modular group]]. The Picard-FuchsPicard–Fuchs equation is then

:<math>\frac{d^2y}{dj^2} + \frac{1}{j} \frac{dy}{dj} +

\frac{31j -4}{144j^2(1-j)^2} y=0.\,</math>

Written in [[Schwarzian derivative|Q-form]], one has

:<math>\frac{d^2f}{dj^2} +

\frac{1-1968j + 2654208j^2}{4j^2 (1-1728j)^2} f=0.\,</math>

==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 [[half-period ratio|period ratio]] &tau;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-FuchsPicard–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]]s. One has

:<math>y(j)=P \left\{ \begin{matrix}

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{1/6} & {1/4} & 0 & j \\

{-1/6\;} & {3/4} & 0 & \;

\end{matrix} \right\}\,</math>

At least four methods to find the [[J-invariant#Inverse and special values|j-function inverse]] can be given.

and so

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:

:<math>\tau(j)=s (\mbox{unfinished, article in development}).</math>

:<math>2(S\tau)

==Identities==

(j) = \frac{1-\frac{1}{4}}{(1-j)^2} + \frac{1-\frac{1}{9}}{j^2} + \frac{1-\frac{1}{4}-\frac{1}{9}}{j(1-j)} = \frac{3}{4(1-j)^2} + \frac{8}{9j^2} + \frac{23}{36j(1-j)}</math>

This solution satisfies the differential equation

where (Sf''Sƒ'')(''x'') is the [[Schwarzian derivative]] of ''fƒ'' with respect to ''x''.

:<math>(S\tau)(j)=\frac{3}{8(1-j)}+\frac{4}{9j^2}+\frac{23}{72j(1-j)}</math>

where (Sf)(x) is the [[Schwarzian derivative]] of ''f'' 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-&ndash;Manin connection]].

==References==

===Pedagogical===

* {{ Citation| last=Schnell | first=Christian | title=On Computing Picard-Fuchs Equations | url=https://www.math.stonybrook.edu/~cschnell/pdf/notes/picardfuchs.pdf }}

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

===References===

* J. Harnad and J. McKay, ''Modular solutions to equations of generalized Halphen type'',

* J. Harnad, ''Picard–Fuchs Equations, Hauptmoduls and Integrable Systems'', Chapter 8 (Pgs. 137&ndash;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|arXiv:solv-int/9902013]]

Proc. R. Soc. London A '''456''' (2000), 261-294

* For a detailed proof of the Picard-Fuchs equation: {{Citation | last1=Milla | first1=Lorenz | title= A detailed proof of the Chudnovsky formula with means of basic complex analysis | arxiv=1809.00533 | year=2018 }}

''(Provides a very readable introduction, some history, references, and various interesting identities

and relations between algebraically related solutions)''

* 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))

''(Provides further examples of Picard-Fuchs equations satisfied by modular function

of genus 0, including non-triangular ones, as well as treating "Inhomogeneous Picard-Fuchs

equations" as special solutions to isomonodromic deformations equations of Painleve type.)''

[[de:j-Funktion]]

{{DEFAULTSORT:Picard-Fuchs Equation}}

[[Category:Elliptic functions]]

[[Category:Modular forms]]