Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [FGL14] Francesc Fité, Josep González, Joan-Carles Lario, Frobenius distribution for quotients of Fermat curves of prime exponent, Canad. J. Math, to appear.
  • [Gre80] Ralph Greenberg, On the Jacobian variety of some algebraic curves, Compositio Math. 42 (1980/81), no. 3, 345-359. MR 607375 (82j:14036)
  • [IK04] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214 (2005h:11005)
  • [KR78] Neal Koblitz and David Rohrlich, Simple factors in the Jacobian of a Fermat curve, Canad. J. Math. 30 (1978), no. 6, 1183-1205. MR 511556 (80d:14022), https://doi.org/10.4153/CJM-1978-099-6

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11G20, 11T24

Retrieve articles in all journals with MSC (2010): 11G20, 11T24


Additional Information

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

American Mathematical Society