On the singularity of the Demjanenko matrix of quotients of Fermat curves

Abstract

Given a prime $\ell \ge 3$ and a positive integer $k\le \ell -2$, one can define a matrix $D_{k,\ell }$, the so-called Demjanenko matrix, whose rank is equal to the dimension of the Hodge group of the Jacobian $\mathrm{J}ac({\mathcal{C}}_{k,\ell })$ of a certain quotient of the Fermat curve of exponent $\ell$. For a fixed $\ell$, the existence of $k$ for which $D_{k,\ell }$ is singular (equivalently, for which the rank of the Hodge group of $\mathrm{J}ac({\mathcal{C}}_{k,\ell })$ is not maximal) has been extensively studied in the literature. We provide an asymptotic formula for the number of such $k$ when $\ell$ tends to infinity.