Low-dimensional factors of superelliptic Jacobians

TL;DR

This paper investigates the structure of Jacobians of superelliptic curves, proving finiteness and boundedness results for abelian varieties of bounded dimension appearing in these Jacobians.

Contribution

It establishes the finiteness and bounded multiplicities of abelian varieties of bounded dimension within the Jacobians of superelliptic curves.

Findings

Finiteness of abelian varieties up to a given dimension in Jacobians.

Bounded multiplicities of these abelian varieties.

Results hold for all superelliptic Jacobians with varying degrees.

Abstract

Given a polynomial $f\in\mathbb{C}[x]$, we consider the family of superelliptic curves $y^d=f(x)$ and their Jacobians $J_d$ for varying integers $d$. We show that for any integer $g$ the number of abelian varieties up to isogeny of dimension $\le g$ which appear in any $J_d$ is finite and their multiplicities are bounded.