Hyperelliptic tangential covers and even elliptic finite-gap potentials, back and forth
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2025
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| _version_ | 1866916586403659776 |
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| author | Treibich, Armando |
| author_facet | Treibich, Armando |
| contents | 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 $\mathcal{P}ot_X(α,d)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16483 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hyperelliptic tangential covers and even elliptic finite-gap potentials, back and forth Treibich, Armando Algebraic Geometry Primary 14H52, 35Q53, 14E20, Secondary 14H40, 14H45, 35Q51, 37K10 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 $\mathcal{P}ot_X(α,d)$. |
| title | Hyperelliptic tangential covers and even elliptic finite-gap potentials, back and forth |
| topic | Algebraic Geometry Primary 14H52, 35Q53, 14E20, Secondary 14H40, 14H45, 35Q51, 37K10 |
| url | https://arxiv.org/abs/2501.16483 |