The Lie Group Structure of Elliptic/Hyperelliptic $\mathrm{\wp }$ Functions

We consider the generalized dual transformation for elliptic/hyperelliptic $\mathrm{\wp }$ functions up to genus three. For the genus one case, from the algebraic addition formula, we deduce that the Weierstrass $\mathrm{\wp }$ function has the SO(2,1) $\cong$ Sp(2,$\mathbb{R}$)/${\mathbb{Z}}_2$ Lie group structure. For the genus two case, by constructing a quadratic invariant form, we find that hyperelliptic $\mathrm{\wp }$ functions have the SO(3,2) $\cong$ Sp(4,$\mathbb{R}$)/${\mathbb{Z}}_2$ Lie group structure. Making use of quadratic invariant forms reveals that hyperelliptic $\mathrm{\wp }$ functions with genus three have the SO(9,6) Lie group and/or it’s subgroup structure.