Elliptic gamma function - Revision history

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2014-08-13T07:24:25Z

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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~~.		I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. To climb is something I really appreciate doing. Some time in the past he selected to reside in North Carolina and he doesn't strategy on changing it. He is an information officer.<br><br>Feel free to visit my blog post real psychic ([http://www.barloof.com/users/MWolfgram relevant site www.barloof.com])

	~~==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 [[modular invariant]]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 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~~

	~~:<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]] 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]]s. One has~~

	:~~<math>y(j)=P \left\{ \begin{matrix}~~
	~~0 & 1 & \infty & \; \\~~
	{1/~~6} & {1~~/~~4} & 0 & j \\~~
	~~{-1/6\;} & {3/4} & 0 & \;~~
	~~\end{matrix} \right\}\,</math>~~

	~~For an explicit formula of an inverse of the ''j''-invariant see the article listed first in the references.~~

	~~Dedekind defines the ''j''-fn by its Schwarz derivative in his letter to Borchardt~~. ~~As a partial fraction, it reveals the geometry of the~~
	~~fundamental domain: Here the first term is in error~~. ~~We should see:~~

	~~: <math> 2S\tau(j) = (1-1/4)/(1-j)^2 + (1-1~~/~~9)/j^2 + (1-1/4-1/9)/j(1-j). \, </math>~~

	~~==Identities==~~
	~~This solution satisfies the differential equation~~

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

	~~where (''Sƒ'')(''x'') is the [[Schwarzian derivative]] 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==~~
	* {{cite arXiv \|last=Adlaj \|first=Semjon \|eprint=1110.3274 \|class=math.NT \|title=An inverse of the modular invariant \|year=2011}}

	* [[J. Harnad]~~] and J. McKay, ''Modular solutions to equations of generalized Halphen type'', Proc. R. Soc. London A '''456''' (2000~~)~~, 261–294,~~
	~~:(Provides a readable introduction, some history, references, and various interesting identities and relations between 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 functions of genus 0, including non-triangular ones, and introduces ''Inhomogeneous Picard–Fuchs equations'' as special solutions to [[isomonodromic deformation]] equations of [[Painlevé type]].)

	~~{{DEFAULTSORT:Picard-Fuchs Equation}}~~
	~~[[Category:Elliptic functions]]~~
	~~[[Category:Modular forms]]~~
	~~[[Category:Hypergeometric functions]]~~
	~~[[Category:Ordinary differential equations]]~~

	~~[[de:J-Funktion]]~~

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← Older revision		Revision as of 11:49, 28 February 2013
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	~~Hi there~~, ~~I am Alyson Boon although it~~ is ~~not~~ the ~~title on my birth certification~~. ~~What me~~ and ~~my family members love~~ is ~~doing ballet but I~~'~~ve been using~~ on ~~new issues recently~~. ~~I am presently~~ a ~~travel agent~~. ~~My spouse~~ and ~~I live~~ in ~~Mississippi and I adore every day living here~~.<br><br>~~my blog~~ :: ~~certified psychics~~ (~~[http:~~//~~Thebankchina~~.~~com~~/~~index.php?do~~=~~/profile~~-~~12480~~/~~info/ click on Thebankchina~~.~~com Thebankchina~~.~~com~~])		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.

			==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 [[modular invariant]]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 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

			:<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]] 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]]s. One has

			:<math>y(j)=P \left\{ \begin{matrix}
			0 & 1 & \infty & \; \\
			{1/6} & {1/4} & 0 & j \\
			{-1/6\;} & {3/4} & 0 & \;
			\end{matrix} \right\}\,</math>

			For an explicit formula of an inverse of the ''j''-invariant see the article listed first in the references.

			Dedekind defines the ''j''-fn by its Schwarz derivative in his letter to Borchardt. As a partial fraction, it reveals the geometry of the
			fundamental domain: Here the first term is in error. We should see:

			: <math> 2S\tau(j) = (1-1/4)/(1-j)^2 + (1-1/9)/j^2 + (1-1/4-1/9)/j(1-j). \, </math>

			==Identities==
			This solution satisfies the differential equation

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

			where (''Sƒ'')(''x'') is the [[Schwarzian derivative]] 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==
			* {{cite arXiv \|last=Adlaj \|first=Semjon \|eprint=1110.3274 \|class=math.NT \|title=An inverse of the modular invariant \|year=2011}}

			* [[J. Harnad]] and J. McKay, ''Modular solutions to equations of generalized Halphen type'', Proc. R. Soc. London A '''456''' (2000), 261–294,
			:(Provides a readable introduction, some history, references, and various interesting identities and relations between 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 functions of genus 0, including non-triangular ones, and introduces ''Inhomogeneous Picard–Fuchs equations'' as special solutions to [[isomonodromic deformation]] equations of [[Painlevé type]].)

			{{DEFAULTSORT:Picard-Fuchs Equation}}
			[[Category:Elliptic functions]]
			[[Category:Modular forms]]
			[[Category:Hypergeometric functions]]
			[[Category:Ordinary differential equations]]

			[[de:J-Funktion]]

71.231.177.16: Attempt to fix math display error

2011-10-24T10:20:58Z

Attempt to fix math display error

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