Hodge groups of certain 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 $Hdg.Inthispaperwestudythesemisimplepart(commutatorsubgroup)of$Hdg$.Assumingthat$p$doesnotdivide$n$and$n-1$isnotdivisibleby$q$,theGaloisgroupof$f(x)$over$K$iseitherthefullsymmetricgroup$S_n$orthealternatinggroup$A_n$,weprovethatthesemisimplepartof$Hdg$ is "as large as possible".

arXiv comment18 pages, the paper will appear in Math. Research Letters