Period Relations for Quaternionic Elliptic Functions

Abstract

We study elliptic functions in quaternionic analysis, and prove some analogues of classical theorems from the complex case. The main result is a relation between the periods of closed differential 1-forms and 3-forms on H/L where L is a lattice in H. By applying to forms that are regular in the sense of quaternionic analysis, we obtain quaternionic analogues of Riemann's period relations for an abelian surface. We also obtain a quaternionic analogue of Legendre's relation, as an equality between $2\pi^2$ and a linear combination of the four quasi-periods of the quaternionic Weierstrass $\zeta$-function, where the coefficients are rational functions of a basis for L. When L is a left-ideal for a maximal order in a definite quaternion algebra over Q, we obtain three further relations among the quasi-periods.