The relative class number one problem for function fields, I

We reduce the classification of finite extensions of function fields (of curves over finite fields) with the same class number to a finite computation; complete this computation in all cases except when both curves have base field $\mathbb{F}_2$ and genus $>1$; and give a conjectural answer in the remaining cases. The conjecture will be resolved in subsequent papers.Comment: 17 pages, including 3 pages of tables; associated code available at https://github.com/kedlaya/same-class-number/; v2: refereed versio