Periods and Quasiperiods

Abstract

In the previous chapter, we proved that the fundamental periods ω ₁, ω ₂ of a Weierstrass $\wp$-function whose corresponding g ₂, g ₃ are algebraic are necessarily transcendental. A similar question arises for the nature of the associated quasi-periods η ₁, η ₂. We shall show that these are also transcendental whenever g ₂ and g ₃ are algebraic. To this end, we shall need the following lemmas. Let $\mathbb{H}$ denote the upper half-plane, i.e. the set of complex numbers z with ℑ(z) > 0.