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.

Let

be the j-invariant with ${\displaystyle g_{2}}$ and ${\displaystyle g_{3}}$ the modular invariants of the elliptic curve in Weierstrass form:

Note that the j-invariant is an isomorphism from the Riemann surface ${\displaystyle \mathbb {H} /\Gamma }$ to the Riemann sphere ${\displaystyle \mathbb {C} \cup \{\infty \}}$; where ${\displaystyle \mathbb {H} }$ is the upper half-plane and ${\displaystyle \Gamma }$ is the modular group. The Picard–Fuchs equation is then

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