  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.
