# Picard–Fuchs equation

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 curves.

## Definition

Let

be the j-invariant with and the modular invariants of the elliptic curve in Weierstrass form:

Note that the *j*-invariant is an isomorphism from the Riemann surface to the Riemann sphere ; where is the upper half-plane and is the modular group. The Picard–Fuchs equation is then

Written in Q-form, one has

## 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-functions. One has

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:

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

- Adlaj, Semjon (2011). "An inverse of the modular invariant". arXiv:1110.3274 [math.NT].
- 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.)