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

TL;DR

This paper investigates the conditions under which the Demjanenko matrix associated with quotients of Fermat curves becomes singular, providing an asymptotic count of such cases as the prime exponent grows large.

Contribution

It offers an asymptotic formula for the number of quotients where the Demjanenko matrix is singular, advancing understanding of the Hodge group dimensions in these cases.

Findings

Asymptotic count of singular Demjanenko matrices for large primes

Characterization of when the Hodge group dimension is not maximal

Extension of previous studies on Fermat curve quotients

Abstract

Given a prime $\ell\geq 3$ 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 ${\mathrm Jac}({\mathcal C}_{k,\ell})$ 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 ${\mathrm Jac}({\mathcal C}_{k,\ell})$ 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.