Hodge Groups of certain Superelliptic Jacobians II

TL;DR

This paper determines the Hodge group of simple factors of Jacobians of certain superelliptic curves defined over fields of characteristic zero, expanding understanding of their algebraic and geometric properties.

Contribution

It provides a detailed analysis of the Hodge groups for Jacobians of superelliptic curves y^q=f(x), where q is a prime power, for the first time in this context.

Findings

Hodge groups are explicitly characterized for these Jacobians.

Results apply to superelliptic curves with polynomial f(x) without multiple roots.

The work advances the classification of algebraic groups associated with superelliptic Jacobians.

Abstract

Let K be a field of characteristic zero, f(x) be a polynomial with coefficients in K and without multiple roots. We consider the superelliptic curve C_{f,q} defined by y^q=f(x), where q=p^r is a power of a prime p. We determine the Hodge group of the simple factors of the Jacobian of C_{f,q}.