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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the singularity of the Demjanenko matrix of quotients of Fermat curves
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by Francesc Fité and Igor E. Shparlinski PDF
Proc. Amer. Math. Soc. 144 (2016), 55-63 Request permission

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.
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Additional Information
  • Francesc Fité
  • Affiliation: Institut für Experimentelle Mathematik/Fakultät für Mathematik, Universität Duisburg-Essen, D-45127 Essen, Germany
  • MR Author ID: 995332
  • Email: francesc.fite@gmail.com
  • Igor E. Shparlinski
  • Affiliation: Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
  • MR Author ID: 192194
  • Email: igor.shparlinski@unsw.edu.au
  • Received by editor(s): June 14, 2014
  • Received by editor(s) in revised form: December 10, 2014
  • Published electronically: July 1, 2015
  • Communicated by: Matthew A. Papanikolas
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 55-63
  • MSC (2010): Primary 11G20, 11T24
  • DOI: https://doi.org/10.1090/proc12717
  • MathSciNet review: 3415576