GAAG seminar - The relative class number one problem for function fields

For any finite extension of function fields of curves over 
finite fields, the relative class number is a positive integer. The 
question of classifying all cases where this integer equals 1 was first 
discussed by Leitzel and Madan in 1976, but seems to have received 
little attention subsequently. We survey our recent solution of this 
problem, which shares many ingredients with the literature on bounding 
the maximal number of points on a curve of fixed genus over a fixed 
finite fields. Among these are the use of exhaustive searches for Weil 
polynomials (a/k/a "Lauter's method"); "linear programming" bounds based 
on Weil's explicit formula; and some character-theoretic arguments to 
rule out noncyclic Galois groups. We also discuss the prospect of 
establishing effective lower bounds on relative class numbers in general.