On the singularity of the demjanenko matrix of quotients of fermat curves

Francesc Fité, Igor E. Shparlinski

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1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)55-63
Number of pages9
JournalProceedings of the American Mathematical Society
Volume144
Issue number1
DOIs
Publication statusPublished - 1 Jan 2016
Externally publishedYes

Keywords

  • Demjanenko matrix
  • Fermat curve
  • Sato-tate conjecture

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