On the Number of Rational Points on Special Families of Curves over Function Fields


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New Zealand Journal of Mathematics

Vol. 47, (2017), Pages 1-7

Douglas Ulmer

School of Mathematics,

Georgia Institute of Technology,

Atlanta, GA 30332, USA


José Felipe Voloch

School of Mathematics and Statistics,

University of Canterbury,

Private Bag 4800, Christchurch 8140, New Zealand}


Abstract We construct families of curves which provide counterexamples for a uniform boundedness question. These families generalize those studied previously by several authors in [Ulm14b], [BHP+15], and [CUV12]. We show, in detail, what fails in the argument of Caporaso, Harris, Mazur that uniform boundedness follows from the Lang conjecture. We also give a direct proof that these curves have finitely many rational points and give explicit bounds for the heights and number of such points.

Keywords algebraic curves, function fields, rational points.

Classification (MSC2000) 11G30.

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