arXiv2026
The Jacobi elliptic functions $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ and their ratios are expressed by Weierstrass's functions $\mathrm{al}_a=\sqrt{x-b_a}$ of an elliptic curve. For a hyperelliptic curve of genus $g$, Weierstrass's $\mathrm{al}_a$ functions generalize $\mathrm{sn}$, $\mathrm{cn}$ and $\mathrm{dn}$, and the modified Korteweg-de Vries and sine-Gordon equations stand behind them. In this paper we study the $\mathrm{al}_{ab}$ functions, which are defined by the sigma function shifted by the sum of two half-periods and which, together with the ratios of the $\mathrm{al}_a$ functions, generalize the remaining Jacobi elliptic functions. We determine the exact lattice of periods of $\mathrm{al}_{12}$ and show that it is single-valued on the unramified double covering of the Jacobian corresponding to the double covering $\hat{w}^2=(x-b_1)(x-b_2)$ of the curve. We show that $\mathrm{al}_{12}$ is the Wronskian of the band-edge eigenfunctions $\mathrm{al}_1$ and $\mathrm{al}_2$, so that the half-period translations realize Darboux transformations. As the main result, $\mathrm{al}_{12}$, $\mathrm{al}_2/\mathrm{al}_1$ and $1/\mathrm{al}_{12}$ satisfy one and the same degenerate Krichever-Novikov equation, whose polynomial has the double roots $0$ and $\infty$, and the half-periods act on these solutions as the Möbius symmetries of the equation. Thus the Krichever-Novikov equation stands behind the $\mathrm{al}_{ab}$ functions. In genus one these functions reduce to $\mathrm{sn}$, $\mathrm{ns}$ and $\mathrm{dc}$.