\'Equation de Pell-Abel et applications

TL;DR

This paper investigates solutions to the Pell-Abel equation on hyperelliptic curves, establishing conditions for existence based on degree and genus, and explores related differential and point order properties.

Contribution

It provides new existence criteria for solutions of Pell-Abel equations and related structures on hyperelliptic curves, revealing previously unknown implications.

Findings

Solutions exist for degree r if and only if r > g.

Existence of primitive k-differentials with a unique zero of order k(2g-2).

Non-Weierstrass points of order n exist if and only if n > 2g.

Abstract

In this paper, we show that there are solutions of every degree $r$ of the equation of Pell-Abel on some real hyperelliptic curve of genus $g$ if and only if $ r > g$. This result, which is known to the experts, has consequences, which seem to be unknown to the experts. First, we deduce the existence of a primitive $k$-differential on an hyperelliptic curve of genus $g$ with a unique zero of order $k(2g-2)$ for every $(k,g)\neq(2,2)$. Moreover, we show that there exists a non Weierstrass point of order $n$ modulo a Weierstrass point on a hyperelliptic curve of genus $g$ if and only if $n > 2g$.