Non-isogenous superelliptic jacobians II

Abstract: Let $\ell$ be an odd prime and $K$ a field of characteristic different from $\ell$. Let $\bar{K}$ be an algebraic closure of $K$. Assume that $K$ contains a primitive $\ell$th root of unity. Let $n \ne \ell$ be another odd prime. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider superelliptic curves $C_{f,\ell}: y^{\ell}=f(x)$ and $C_{h,\ell}: y^{\ell}=h(x)$ of genus $(n-1)(\ell-1)/2$, and their jacobians $J^{(f,\ell)}$ and $J^{(h,\ell)}$, which are $(n-1)(\ell-1)/2$-dimensional abelian varieties over $\bar{K}$. Suppose that one of the polynomials is irreducible and the other reducible over $K$. We prove that if $J^{(f,\ell)}$ and $J^{(h,\ell)}$ are isogenous over $\bar{K}$ then both endomorphism algebras $\mathrm{End}^{0}(J^{(f,\ell)})$ and $\mathrm{End}^{0}(J^{(h,\ell)})$ contain an invertible element of multiplicative order $n$.