- Picard-Fuchs equation
In
mathematics , the Picard-Fuchs equation is a linearordinary differential equation whose solutions describe the periods ofelliptic curve s.Definition
Let
:j=frac{g_2^3}{g_2^3-27g_3^2}
be the
j-invariant with g_2 and g_3 themodular invariant s of the elliptic curve in Weierstrass form::y^2=4x^3-g_2x-g_3.,
Note that the j-invariant is an
isomorphism from theRiemann surface H/ Gamma to theRiemann sphere mathbb{C}cup{infty}; where H is theupper half-plane and Γ is themodular group . The Picard-Fuchs equation is then:frac{d^2y}{dj^2} + frac{1}{j} frac{dy}{dj} + frac{31j -4}{144j^2(1-j)^2} y=0.,
Written in Q-form, one has
:frac{d^2f}{dj^2} + frac{1-1968j + 2654208j^2}{4j^2 (1-1728j)^2} f=0,
olutions
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 aSchwarz 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 ofRiemann P-function s. One has:y(j)=P left{ egin{matrix} 0 & 1 & infty & ; \ {1/6} & {1/4} & 0 & j \{-1/6;} & {3/4} & 0 & ;end{matrix} ight},
and so
:au(j)=s (mbox{unfinished, article in development}).
Identities
This solution satisfies the differential equation
:S au)(j)=frac{3}{8(1-j)}+frac{4}{9j^2}+frac{23}{72j(1-j)}
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, theGauss-Manin connection .References
*
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 toisomonodromic deformation equations ofPainlevé type .)
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