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Ranks in Families of Jacobian Varieties of Twisted Fermat Curves

  Published:2009-12-04
 Printed: Mar 2010
  • Andrzej Dąbrowski,
    University of Szczecin, Institute of Mathematics, ul. Wielkopolska 15, 70-451 Szczecin, Poland
  • Tomasz Jędrzejak,
    University of Szczecin, Institute of Mathematics, ul. Wielkopolska 15, 70-451 Szczecin, Poland
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Abstract

In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.
Keywords: Fermat curve, Jacobian variety, elliptic curve, canonical height Fermat curve, Jacobian variety, elliptic curve, canonical height
MSC Classifications: 11G10, 11G05, 11G50, 14G05, 11G30, 14H45, 14K15 show english descriptions Abelian varieties of dimension $> 1$ [See also 14Kxx]
Elliptic curves over global fields [See also 14H52]
Heights [See also 14G40, 37P30]
Rational points
Curves of arbitrary genus or genus $
Special curves and curves of low genus
Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]
11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]
11G05 - Elliptic curves over global fields [See also 14H52]
11G50 - Heights [See also 14G40, 37P30]
14G05 - Rational points
11G30 - Curves of arbitrary genus or genus $
14H45 - Special curves and curves of low genus
14K15 - Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]
 

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