Classification of algebraic function fields with class number one

Abstract

In this paper we prove that there are exactly eight function fields, up to isomorphism, over finite fields with class number one and positive genus. This classification was already suggested, although not completely proved, in a previous work about this topic

Maps between curves and arithmetic obstructions

“…There are exactly 8 curves for which H(X) = X, equivalently, curves whose function fields have class number one [14,8]. We have verified that Conjecture 2.2 holds when X and Y are among this list of 8 curves, and it therefore holds if either H(X) = X or H(Y ) = Y .…”

Function fields of class number one

“…We are submitting our independent work, cutting out unnecessary duplication with the work of Mercuri and Stirpe. It should be noted that [MSp14] offers two proofs, the second being essentially the same as ours.…”

“…But in checking all the possible candidate cases, they had discarded wrongly the unique correct case, as we found out and as also pointed out in [MSp14]. To avoid the duplication, we refer to the end of [MSp14] for details and notation, and content ourselves with only a few relevant comments.…”

“…In the end of [MSp14], the authors gave a table of rational points of curves defined by C i and Q i + L(k 1 , k 2 , k 3 , k 4 ) 2 , i = 1, 2, 3, 4, where…”

“…After informing Stirpe of this, we were told by him that, jointly with Mercuri, he also had recently succeeded in proving the same result. The preprint [MSp14] has now appeared. We are submitting our independent work, cutting out unnecessary duplication with the work of Mercuri and Stirpe.…”