The al function of a cyclic trigonal curve of genus three

Abstract

A cyclic trigonal curve of genus three is a $\mathbb{Z}_3$ Galois cover of $\mathbb{P}^1$, therefore can be written as a smooth plane curve with equation $y^3 = f(x) =(x - b_1) (x - b_2) (x - b_3) (x - b_4)$. Following Weierstrass for the hyperelliptic case, we define an ``$\mathrm{al}$'' function for this curve and $\mathrm{al}^{(c)}_r$, $c=0,1,2$, for each one of three particular covers of the Jacobian of the curve, and $r=1,2,3,4$ for a finite branchpoint $(b_r,0)$. This generalization of the Jacobi $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ functions satisfies the relation: $$ \sum_{r=1}^4 \frac{\prod_{c=0}^2\mathrm{al}_r^{(c)}(u)}{f'(b_r)} = 1 $$ which generalizes $\mathrm{sn}^2u + \mathrm{cn}^2u = 1$. We also show that this can be viewed as a special case of the Frobenius theta identity.