The al function of a cyclic trigonal curve of genus three

TL;DR

This paper introduces an 'al' function for cyclic trigonal genus three curves, generalizing classical elliptic functions, and demonstrates its relation to Frobenius theta identities.

Contribution

It defines a new 'al' function for cyclic trigonal curves and establishes its fundamental relation, extending classical elliptic function theory to higher genus curves.

Findings

Defines 'al' functions for genus three cyclic trigonal curves

Establishes a key relation generalizing elliptic identities

Connects the 'al' functions to Frobenius theta identities

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.