Elimination of the parameters in an elliptic parametrization of the Schwarz primitive surface

Abstract

Let $Ω$ be the curvilinear square bounded by the four circular arcs $|ζ\mp1|=\sqrt{2}$, $|ζ\mp i|=\sqrt{2}$, and let $W=\sqrt{1+14ζ^{4}+ζ^{8}}$. Put \[ θ(ζ)=\arcsin\frac{2(1+i)ζ} {\sqrt{1+4iζ^{2}-ζ^{4}}}, \] \[ f=\tfrac14\bigl(-iF[θ,\tfrac14]+F[θ,\tfrac34]\bigr),\quad \quad g=\tfrac14\bigl(\ \ iF[θ,\tfrac14]+F[θ,\tfrac34]\bigr), \] \[ \mathfrak h=(2-\sqrt3)\, F\bigl[\arcsin\bigl(i(2+\sqrt3)ζ^{2}\bigr),(2-\sqrt3)^{4}\bigr]. \] These are, verbatim, the three functions of our companion paper $\mathrm{[D]}$ on the diamond surface $\mathrm{D}$: the two papers start from the same point, and differ only in which real parts are taken. With $X=(κ\,\operatorname{Re}f,\ κ\,\operatorname{Re}g,\ \tfrac12+κ\,\operatorname{Im}\mathfrak{h})$ and $κ=3/(2K[1/9])$, the map $X$ parametrizes a fundamental patch of Schwarz's primitive surface $\mathrm{P}$, the conjugate of $\mathrm{D}$. Part I builds the algebraic apparatus $\ldots$ the normalizing constant is identified exactly: $\varpi:=κ^{-1}=K[-3]=\tfrac12 K[3/4]=\tfrac23 K[1/9]$. Part II carries out the elimination and completes the proof of \[ (\star)\qquad \operatorname{sn}\bigl(\varpi(x{+}y),-3\bigr)\,\operatorname{sn}\bigl(\varpi(x{-}y),-3\bigr) =\tfrac13\operatorname{sn}\bigl(3\varpi(z-\tfrac12),\tfrac19\bigr). \] Part III shows that $(\star)$ is, after all, additively separable $\ldots$ for an overview of this paper and $\mathrm{[D]}$, please see arXiv:2609.14206.