The al function of a cyclic trigonal curve of genus three

الأصل · EN

A cyclic trigonal curve of genus three is a Z₃ Galois cover of P¹, therefore can be written as a smooth plane curve with equation y³ = f(x) =(x - b₁) (x - b₂) (x - b₃) (x - b₄). Following Weierstrass for the hyperelliptic case, we define an ``al'' function for this curve and 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 (bᵣ,0). This generalization of the Jacobi sn, cn, dn functions satisfies the relation: ∑ᵣ₌₁⁴ ∏c₌₀²alᵣ⁽ᶜ⁾(u)f'(bᵣ) = 1 which generalizes sn²u + cn²u = 1. We also show that this can be viewed as a special case of the Frobenius theta identity.