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 ${\mathrm{al}}_r^{(c)}$, $c=0,1,2$, 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: $$\sum _{r=1}^4\frac{{\prod }_{c=0}^2{\mathrm{al}}_r^{(c)}(u)}{f^{\prime }(b_r)}=1$$ which generalizes ${\mathrm{sn}}^{2}u+{\mathrm{cn}}^{2}u=1$. We also show that this can be viewed as a special case of the Frobenius theta identity.

40 pages

References in corpus (1)

• Neumann system and hyperelliptic al functions

Cited by in corpus (1)

• The sigma function over a family of cyclic trigonal curves with a singular fiber