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

### From NZJM

**New Zealand Journal of Mathematics**

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

**Douglas Ulmer**

School of Mathematics,

Georgia Institute of Technology,

Atlanta, GA 30332, USA

mailto:douglas.ulmer@math.gatech.edu

**JosÃ© Felipe Voloch **

School of Mathematics and Statistics,

University of Canterbury,

Private Bag 4800, Christchurch 8140, New Zealand}

mailto:felipe.voloch@canterbury.ac.nz

**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.