The "exponential" torsion of superelliptic Jacobians

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