Elliptic genera

Abstract

Let ω₁, ω₂ ∈ ℂ such that τ:= ω₂/ω₁ ∉ ℝ∪{∞}; we can number ω₁ and ω₂ such that Im (τ) > 0. Then L = ℤ•ω₂ + ℤ•ω₁ is a lattice in ℂ; put L’ = L\{0}. For z ∈ ℂ,

$$ wp(z): = \frac{1}{{{x^2}}} + \sum\limits_{\omega \in L'} {\left( {\frac{1}{{{{(z - \omega )}^2}}} - \frac{1}{{{\omega ^2}}}} \right)} $$

defines a meromorphic function with poles of order two at all lattice points. This function is called the Weierstraß p-function.