The relative class number one problem for function fields, I

Abstract

We classify finite extensions of function fields (of curves over finite fields) with the same class number, except when both curves have base field F 2 and genus > 1. In this case, we show that the zeta functions of the curves form one of 208 explicit pairs, and find 64 examples of cyclic extensions; the list is provably complete except if the base curve has genus 6 or 7 or if the extension is noncyclic. These cases will be resolved in subsequent papers.

The relative class number one problem for function fields, II

“…As a side effect of the analysis for d = 7, we also establish another missing assertion from [4]: there is exactly one curve of genus 6 over F 2 which occurs as a cover of a curve of genus 1 with relative class number 1 (Lemma 10.2). This confirms the corresponding entry in [4, Table 3].…”

“…This paper continues the work done in [4] on the relative class number one problem for function fields, building upon work of Leitzel-Madan [7] and Leitzel-Madan-Queen [8]. That is, we seek to identify finite extensions F ′ /F of function fields of curves over finite fields for which the two class numbers are equal.…”

“…That is, we seek to identify finite extensions F ′ /F of function fields of curves over finite fields for which the two class numbers are equal. In this paper, we establish the following result; the argument relies heavily on results from [4] as described below.…”

“…Before continuing, we recall some notation from [4]. Given a finite extension F ′ /F of function fields, we write C, C ′ for the curves corresponding to F, F ′ ; q F , q F ′ for the orders of the base fields of C, C ′ ; g F , g F ′ for the genera of C, C ′ ; and h F , h F ′ for the class numbers of F, F ′ .…”

“…In this paper we only consider the case where q F = q F ′ , in which case we say F ′ /F is a purely geometric extension (the other extreme is when g F ′ = g F , in which case we say F ′ /F is a constant extension). We recall here [4,Theorem 1.3(a)].…”