Classification of Algebraic Function Fields with Divisor Class Number Two

Abstract

In a previous paper we proved that there are 11 quadratic algebraic function fields with divisor class number two. Here we complete the classification of algebraic function fields with divisor class number two giving all non-quadratic solutions. Our result is the following. Let us denote by k the finite field with q elements. Up to isomorphism, there are exactly 8 non-quadratic algebraic function fields of one variable K/k having k for full constant field and with a divisor class number equal to two.

Classification of function fields with class number three

“…In [3], Le Brigand classified the non-quadratic ones with class number two. We summarize these results in Table 3.…”

“…Here we use some important tools from algebraic geometry, in particular the canonical model of the algebraic function field K over the full constant field F q , which is a normal smooth curve of degree 2g-2 in P g−1 (F q ), where g is the genus of K. This method was already used in [3]. In order to complete the list and non-isomorphism of the mutual algebraic function fields over the full constant field, we also use the computer algebra package MAGMA in [2].…”

“…The classification of the algebraic function fields with class number two was completed by Le Brigand in [4] and [3]. In [4], up to isomorphism, she classified all quadratic algebraic function fields K with h K = 2.…”

“…by the proof in Section 5.2.1 of[3]. Up to isomorphism with respect to M , we get 2 equations for the cubic surface given in Appendix A.Let U be the open set of P 3 (k) defined by T = 0.…”

“…Appendix A. Non-hyperelliptic function fields of genus 4 with class number three When K is a non-hyperelliptic function field of genus 4, the canonical model C is the intersection of a quadric surface D and a cubic surface S in the projective 3-space over the finite field of 2 elements by Lemma 5.1 of [3]. Moreover, D is a hyperbolic quadric, cone or an elliptic quadric.…”