Hodge groups of certain superelliptic jacobians

TL;DR

This paper investigates the Hodge group of superelliptic Jacobians, proving that under certain conditions, its semisimple part is maximally large, extending previous work on its center.

Contribution

It establishes the maximal size of the semisimple part of the Hodge group for specific superelliptic Jacobians, building on prior analysis of its center.

Findings

Semisimple part of Hodge group is maximally large under given conditions.

Galois group of polynomial is either full symmetric or alternating group.

Results extend understanding of Hodge groups for superelliptic Jacobians.

Abstract

Suppose that $K$ is a field of characteristic 0, $p$ is an odd prime, $r$ a positive integer, $q=p^r$ a prime power. Suppose that $f(x)$ is a polynomial of degree $n > 4$ with coefficients in $K$ and without multiple roots. Let us consider the superelliptic curve $C: y^q=f(x)$ and its jacobian $J(C)$. Assuming that $K$ is a subfield of the field of complex numbers, we study the (connected reductive algebraic) Hodge group $Hdg$ of the corresponding complex abelian variety $J(C)$. In our previous paper (arXiv:0907.1563 [math.AG]) we studied the center of $Hdg. In this paper we study the semisimple part (commutator subgroup) of $Hdg$. Assuming that $p$ does not divide $n$ and $n-1$ is not divisible by $q$, the Galois group of $f(x)$ over $K$ is either the full symmetric group $S_n$ or the alternating group $A_n$, we prove that the semisimple part of $Hdg$ is "as large as possible".