Hyperelliptic tangential covers and even elliptic finite-gap potentials, back and forth
A. Treibich Departamento de Matemática y Aplicaciones, Centro Universitario Regional del Este, Universidad de la República, Montevideo, Uruguay
Abstract:
Let
$\pi\colon \Gamma \to X$ denote a degree
$n$ cover of an elliptic curve, marked at a smooth point
$p\in \Gamma$. Consider the (rational) Abel map $\mathrm{Ab}_p\colon \Gamma \to \operatorname{Jac}\Gamma$ and the dual map $\pi^\vee:= \mathrm{Ab}_p\circ\pi^* \colon X \to \operatorname{Jac}\Gamma$ into the Jacobian variety of
$\Gamma$. We then call
$\pi$ a hyperelliptic
tangential cover (ht-cover) if and only if
$\Gamma$ is a hyperelliptic curve,
$p\in \Gamma$ is a Weierstrass point and the images of
$\Gamma$ and
$X$ in
$\operatorname{Jac}\Gamma$ are tangent at its origin. To any such ht-cover
$\pi$ we attach an integer vector
$\mu \in \mathbb{N}^4$, the so-called type, satisfying $\mu_0+1\equiv \mu_1 \equiv \mu_2 \equiv \mu_3 \equiv n \mod2$ and
$2n+1-\sum_i \mu_i^2=4d$, for some
$d\in \mathbb{N}$. Whenever
$\Gamma$ is smooth, the type
$\mu$ gives the number of Weierstrass points of
$\Gamma$ (different from
$p$) over each half-period
$\omega_i$ of
$X$ (
$i=0,\dots,3$). We denote by
$\mathcal{S}\mathcal{C}_X(\mu,d)$ the set of degree
$n$ ht-covers of type
$\mu$. Then the even, doubly-periodic, finite-gap potentials
associated to $\mathcal{S}\mathcal{D}_X(\mu,d):=\{(\pi,\xi)\colon\pi\in\mathcal{S}\mathcal{C}_X(\mu,d),\, \xi\text{ is a theta-characteristic of }\pi\}$ decompose as
$$
u_\xi(x)=\sum_0^3\alpha_i(\alpha_i+1)\wp(x-\omega_i)+2\sum_{j=1}^m \bigl(\wp(x-\rho_j)+\wp(x+\rho_j)\bigr)
$$
for some
$(\alpha,m)\in \mathbb{N}^4\times \mathbb{N}$ such that
$\sum_i\alpha_i(\alpha_i+1)+4m=\sum_i\mu_i^2 +4d$.
The set of such potentials, denoted by
$\mathcal{P}ot_X(\alpha,m)$, is finite, and we have a bijection
$$
(\pi,\xi) \in \bigcup_{(\mu,d)}\mathcal{S}\mathcal{D}_X(\mu,d)\mapsto u_\xi \in \bigcup_{(\alpha,m)}\mathcal{P}ot_X(\alpha,m)\,.
$$
The problem at stake is to find its inverse map, as well as the cardinals
$\#\mathcal{S}\mathcal{C}_X(\mu,d)$ and
$\#\mathcal{P}ot_X(\alpha,m)$. The latter problem has been studied thoroughly for
$\mathcal{P}ot_X(\alpha,0)$ and
$\mathcal{P}ot_X(\alpha,1)$ (for any
$\alpha \in \mathbb{N}^4$). In this article we prove that
$\#\mathcal{P}ot_X(\alpha,2)=27$ for a generic elliptic curve
$X$ and find the inverse image of
$\mathcal{P}ot_X(\alpha,m)$. Bounds for the types and arithmetic genera of their spectral data follow. We conclude
with a conjectural recursive formula in
$d\in \mathbb{N}$ for
$\#\mathcal{P}ot_X(\alpha,d)$ and
$\#\mathcal{S}\mathcal{C}_X(\mu,d)$.
This article is dedicated to the memory of Jean Louis Verdier and Igor M. Krichever.
Bibliography: 30 titles.
Keywords:
even elliptic finite-gap potential, hyperelliptic tangential cover, theta-characteristic, weak del Pezzo surface, blowing-up of a point.
MSC: Primary
14H52,
14H55,
37J38; Secondary
14E20,
35Q53 Received: 18.04.2024 and 10.12.2024
DOI:
10.4213/sm10108
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