Unboundedness of the number of rational points on curves over function   fields
  We give examples of sequences of smooth non-isotrivial curves for every genus at least two, defined over a rational function field of positive characteristic, such that the (finite) number of rational points of the curves in the sequence cannot be uniformly bounded. 
Ricardo Concei\c{c}\~ao, Douglas Ulmer, Jos\'e Felipe Voloch
math.NT