Hyperelliptic tangential covers and even elliptic finite-gap potentials, back and forth
Let $(X,\omega_0):=(\mathbb{C}/\Lambda,0)$ denote the elliptic curve associated to the lattice $\Lambda$, $X_2:=\{\omega_0,\cdots, \omega_3\}$ its set of half-periods and $\wp:X \to \mathbb{P}^1$ the usual Weierstrass $\wp$ function, with a double pole at the origin $\omega_0$. Fix $(\alpha,m)\in \mathbb{N}^4\times \mathbb{N}$ and consider a function $$u_\xi(x) = \sum_0^3 \alpha_i(\alpha_i+1)\wp(x\,\textrm{-}\,\omega_i) +2\sum_{j=1}^m \left(\wp(x\, \textrm{-}\, \rho_j)+\wp(x+\rho_j)\right),$$ where $\{\rho_j\} \in (X \setminus X_2)^{(m)}$. The latter is known to be a so-called (even, $\Lambda$-periodic) finite-gap potential, if and only if $\{\rho_j\} $ satisfies the so-called (D-G) square system of equations. We let $\mathcal{P}ot_X(\alpha,m)$ denote the set of such potentials. Any such potential corresponds to a unique spectral data $(\pi,\xi)$, where $\pi: \Gamma \to X$ is a hyperelliptic tangential cover of degree $n:=\frac{1}{2}(\sum_i\alpha_i(\alpha_i+1)+4m)$ and $\xi$ a $\theta$-characteristic of the spectral curve $\Gamma$. The problem at stake is to find out all spectral data of the family $\mathcal{P}ot_X(m) := \bigcup_{\alpha\in \mathbb{N}^4} \mathcal{P}ot_X(\alpha,m),$ for any $m$. The latter problem has been thoroughly studied for $\mathcal{P}ot_X(0)$ and $\mathcal{P}ot_X(1)$. In this article we go one step further, by studying all spectral data of each family $\mathcal{P}ot_X(\alpha,2)$. We find the bound $\#\mathcal{P}ot_X(\alpha,2)\leq 27$, for any $\alpha\in \mathbb{N}^4$, with equality for a generic elliptic curve $X$. We also find a formula for the arithmetic geni of the corresponding spectral curves in terms of $\alpha$, which we generalize to $\mathcal{P}ot_X(\alpha,m)$ for any $m$. At last, we conclude with a natural conjecture, leading to a recursive formula in $d\in \mathbb{N}$, for the cardinals of $\mathcal{P}ot_X(\alpha,d)$.