The function of a cyclic trigonal curve of genus three

A cyclic trigonal curve of genus three is a Galois cover of , therefore can be written as a smooth plane curve with equation . Following Weierstrass for the hyperelliptic case, we define an "" function for this curve and , , for each one of three particular covers of the Jacobian of the curve, and for a finite branchpoint . This generalization of the Jacobi , , functions satisfies the relation: Sigma(4)(r=1) Pi(2)(c=0) al(r)((c)) (u)/f'(b(r)) = 1 which generalizes . We also show that this can be viewed as a special case of the Frobenius theta identity.