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| 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 curve]]s.
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| ==Definition==
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| Let
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| :<math>j=\frac{g_2^3}{g_2^3-27g_3^2}</math>
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| be the [[j-invariant]] with <math>g_2</math> and <math>g_3</math> the [[modular invariant]]s of the elliptic curve in [[Weierstrass equation|Weierstrass form]]:
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| :<math>y^2=4x^3-g_2x-g_3.\,</math>
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| Note that the ''j''-invariant is an [[isomorphism]] from the [[Riemann surface]] <math> \mathbb{H}/\Gamma </math> to the [[Riemann sphere]] <math>\mathbb{C}\cup\{\infty\}</math>; where <math>\mathbb{H}</math> is the [[upper half-plane]] and <math>\Gamma</math> is the [[modular group]]. The Picard–Fuchs equation is then
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| :<math>\frac{d^2y}{dj^2} + \frac{1}{j} \frac{dy}{dj} +
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| \frac{31j -4}{144j^2(1-j)^2} y=0.\,</math>
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| Written in [[Schwarzian derivative|Q-form]], one has
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| :<math>\frac{d^2f}{dj^2} +
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| \frac{1-1968j + 2654208j^2}{4j^2 (1-1728j)^2} f=0.\,</math>
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| ==Solutions==
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| 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]] 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]].
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| 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]]s. One has
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| :<math>y(j)=P \left\{ \begin{matrix} | |
| 0 & 1 & \infty & \; \\
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| {1/6} & {1/4} & 0 & j \\
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| {-1/6\;} & {3/4} & 0 & \;
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| \end{matrix} \right\}\,</math>
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| For an explicit formula of an inverse of the ''j''-invariant see the article listed first in the references.
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| Dedekind defines the ''j''-fn by its Schwarz derivative in his letter to Borchardt. As a partial fraction, it reveals the geometry of the
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| fundamental domain: Here the first term is in error. We should see:
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| : <math> 2S\tau(j) = (1-1/4)/(1-j)^2 + (1-1/9)/j^2 + (1-1/4-1/9)/j(1-j). \, </math>
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| ==Identities==
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| This solution satisfies the differential equation
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| :<math>(S\tau)(j)=\frac{3}{8(1-j)^2}+\frac{4}{9j^2}+\frac{23}{72j(1-j)}</math>
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| where (''Sƒ'')(''x'') is the [[Schwarzian derivative]] of ''ƒ'' with respect to ''x''.
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| ==Generalization==
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| In [[algebraic geometry]] this equation has been shown to be a very special case of a general phenomenon, the [[Gauss–Manin connection]].
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| ==References==
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| * {{cite arXiv |last=Adlaj |first=Semjon |eprint=1110.3274 |class=math.NT |title=An inverse of the modular invariant |year=2011}}
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| * [[J. Harnad]] and J. McKay, ''Modular solutions to equations of generalized Halphen type'', Proc. R. Soc. London A '''456''' (2000), 261–294,
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| :(Provides a readable introduction, some history, references, and various interesting identities and relations between solutions)
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| * 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)).
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| :(Provides further examples of Picard–Fuchs equations satisfied by modular functions of genus 0, including non-triangular ones, and introduces ''Inhomogeneous Picard–Fuchs equations'' as special solutions to [[isomonodromic deformation]] equations of [[Painlevé type]].)
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| {{DEFAULTSORT:Picard-Fuchs Equation}}
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| [[Category:Elliptic functions]]
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| [[Category:Modular forms]]
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| [[Category:Hypergeometric functions]]
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| [[Category:Ordinary differential equations]]
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| [[de:J-Funktion]]
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