Hyperelliptic tangential covers and even elliptic finite-gap potentials, back and forth
Armando Treibich
TL;DR
The paper develops a finite-gap framework for a new family of even elliptic potentials by connecting each potential to spectral data given by a hyperelliptic tangential cover and a theta-characteristic on the spectral curve. It constructs a global geometric model using a family of rational surfaces and degree-2 projections to reduce the counting problem for spectral data to algebro-geometric tangency problems, obtaining a sharp bound of 27 for the number of spectral data in the $m=2$ case and an explicit genus bound. The results unify theta-characteristics with Severi-type curve counts on rational surfaces and culminate in a conjectural recursive structure for the cardinalities of Pot$_X(α,d)$ across $d$, suggesting a deep, lattice-theoretic organization of spectral data across the entire family. The work also provides explicit formulas for the discriminant geometry and shows that, for generic elliptic curves, the spectral data stabilize to a fixed finite set, enabling precise spectral descriptions of the corresponding hyperelliptic potentials. Overall, the paper advances the algebro-geometric understanding of elliptic finite-gap potentials and opens a path to recursive computation of their spectral data.
Abstract
Let $(X,ω_0):=(\mathbb{C}/Λ,0)$ denote the elliptic curve associated to the lattice $Λ$, $X_2:=\{ω_0,\cdots, ω_3\}$ its set of half-periods and $\wp:X \to \mathbb{P}^1$ the usual Weierstrass $\wp$ function, with a double pole at the origin $ω_0$. Fix $(α,m)\in \mathbb{N}^4\times \mathbb{N}$ and consider a function $$u_ξ(x) = \sum_0^3 α_i(α_i+1)\wp(x\,\textrm{-}\,ω_i) +2\sum_{j=1}^m \left(\wp(x\, \textrm{-}\, ρ_j)+\wp(x+ρ_j)\right),$$ where $\{ρ_j\} \in (X \setminus X_2)^{(m)}$. The latter is known to be a so-called (even, $Λ$-periodic) finite-gap potential, if and only if $\{ρ_j\} $ satisfies the so-called (D-G) square system of equations. We let $\mathcal{P}ot_X(α,m)$ denote the set of such potentials. Any such potential corresponds to a unique spectral data $(π,ξ)$, where $π: Γ\to X$ is a hyperelliptic tangential cover of degree $n:=\frac{1}{2}(\sum_iα_i(α_i+1)+4m)$ and $ξ$ a $θ$-characteristic of the spectral curve $Γ$. The problem at stake is to find out all spectral data of the family $\mathcal{P}ot_X(m) := \bigcup_{α\in \mathbb{N}^4} \mathcal{P}ot_X(α,m),$ 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 $\mathcal{P}ot_X(α,2)$. We find the bound $\#\mathcal{P}ot_X(α,2)\leq 27$, for any $α\in \mathbb{N}^4$, 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 $α$, which we generalize to $\mathcal{P}ot_X(α,m)$ 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 $\mathcal{P}ot_X(α,d)$.