Resumen de The \mathrm {al} function of a cyclic trigonal curve of genus three

Shigeki Matsutani, Emma Previato

• 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:

  \begin{aligned} \sum _{r=1}^4 \frac{\prod _{c=0}^2\mathrm {al}_r^{(c)}(u)}{f'(b_r)} = 1 \end{aligned} 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.