Hyperelliptic tangential covers and even elliptic finite-gap potentials, back and forth

TL;DR

This paper characterizes the spectral data of even elliptic finite-gap potentials associated with hyperelliptic tangential covers, extending previous results to potentials with two gaps and proposing a recursive formula for general cases.

Contribution

It provides a bound on the number of spectral data for potentials with two gaps and generalizes formulas for spectral curve genus to all such potentials.

Findings

Bound of at most 27 spectral data points for generic elliptic curves with two gaps.

Derived a formula for the genus of spectral curves in terms of parameters.

Proposed a recursive conjecture for counting spectral data in higher-gap cases.

Abstract

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 $\mathcal{P}ot_X(\alpha,d)$.