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

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Given a prime ℓ≥ 3 and a positive integer k ≤ ℓ-2, one can define a matrix Dₖ,ℓ, the so-called Demjanenko matrix, whose rank is equal to the dimension of the Hodge group of the Jacobian Jac(Cₖ,ℓ) of a certain quotient of the Fermat curve of exponent ℓ. For a fixed ℓ, the existence of k for which Dₖ,ℓ is singular (equivalently, for which the rank of the Hodge group of Jac(Cₖ,ℓ) is not maximal) has been extensively studied in the literature. We provide an asymptotic formula for the number of such k when ℓ tends to infinity.