Let \(\varvec{J}\) be the Jacobian of a superelliptic curve defined by the equation \(\varvec{y}^{\varvec{\ell }} \varvec{= f(x)}\) , where \(\varvec{f}\) is a separable polynomial of degree non-divisible by \(\varvec{\ell }\) . Recall that \(\varvec{J}\) is an abelian variety of dimension \(\varvec{\frac{1}{2}(\ell - 1)(\text {deg} f-1)}\) . In this article we study the exponential (i.e. \(\varvec{\ell }\) -power) torsion of \(\varvec{J}\) . In particular, under some mild conditions on the polynomial \(\varvec{f}\) , we determine the image of the associated \(\varvec{\ell }\) -adic representation up to the determinant. We show also that the image of the determinant is contained in an explicit \({\varvec{\mathbb {Z}}}_{\varvec{\ell }}\) -lattice with a finite index. As an application, we prove new cases of the Hodge, Tate and Mumford–Tate conjectures for generic superelliptic Jacobians of the above type.
