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


Authors: Francesc Fité and Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 144 (2016), 55-63
MSC (2010): Primary 11G20, 11T24
DOI: https://doi.org/10.1090/proc12717
Published electronically: July 1, 2015
MathSciNet review: 3415576
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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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Francesc Fité
Affiliation: Institut für Experimentelle Mathematik/Fakultät für Mathematik, Universität Duisburg-Essen, D-45127 Essen, Germany
Email: francesc.fite@gmail.com

Igor E. Shparlinski
Affiliation: Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
Email: igor.shparlinski@unsw.edu.au

DOI: https://doi.org/10.1090/proc12717
Keywords: Fermat curve, Demjanenko matrix, Sato-Tate conjecture
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
Article copyright: © Copyright 2015 American Mathematical Society