Modular discriminant

In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass. This class of functions are also referred to as p-functions and generally written using the symbol ℘. The ℘ functions constitute branched double coverings of the Riemann sphere by the torus, ramified at four points. They can be used to parametrize elliptic curves over the complex numbers, thus establishing an equivalence to complex tori. They also yield solutions of the Korteweg–de Vries equation.

Symbol for Weierstrass P function

The Weierstrass elliptic function can be defined in three closely related ways, each of which possesses certain advantages. One is as a function of a complex variable z and a lattice Λ in the complex plane. Another is in terms of z and two complex numbers ω₁ and ω₂ defining a pair of generators, or periods, for the lattice. The third is in terms of z and a modulus τ in the upper half-plane. This is related to the previous definition by τ = ω₂/ω₁, which by the conventional choice on the pair of periods is in the upper half-plane. Using this approach, for fixed z the Weierstrass functions become modular functions of τ.

In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods ω₁ and ω₂ defined as

Then ${\displaystyle \Lambda =\{m\omega _{1}+n\omega _{2}:m,n\in \mathbb {Z} \}}$ are the points of the period lattice, so that

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