On the Weil descent of Artin-Schreier algebraic function fields over finite fields
Stéphane Ballet, Robert Rolland
TL;DR
This work advances the study of Weil descent for Artin–Schreier type function fields over finite fields by establishing concrete descent criteria for subextensions of generalized Artin–Schreier fields, and by giving a thorough analysis of the additive-case $A(T)=T^{p^s}+T$. It first lays out a general descent framework in which a field $F$ with full constant field $K$ can descend to a subfield $k$ via a semidirect product structure $G\rtimes \Gamma$, and then specializes to the generalized Artin–Schreier setting where $F/L$ is an elementary abelian $p$-extension generated by additive shifts $y\mapsto y+\alpha$ with $A(\alpha)=0$. Focusing on $A(T)=T^{p^s}+T$, the paper derives explicit descriptions of the group $G$, enumerates its subgroups, and analyzes how Frobenius actions determine which subextensions descend to smaller constant fields, with detailed treatment of the case $s=2$, $t=1$ and several subcases depending on $p$. The authors also provide explicit examples for small primes to illustrate the descent patterns and invariant subgroups, offering practical criteria for constructing curves with prescribed descent properties. Overall, the results yield explicit descent criteria and constructive descriptions that can guide arithmetic and geometric investigations of Artin–Schreier type function fields over finite fields.
Abstract
Let us consider a generalized Artin-Schreier algebraic function field extension $F$ of the rational function field $\F_{p^n}(x)$ defined over the finite field extension $K=\F_{p^n}$ of the prime field $\F_p$. We assume that $K$ is algebraically closed in $F$. We give general results on the descent over the fields $k= \F_{p^t}$ for $t$ dividing $n$. Then, we completely handle the bi-cyclic case of the descent over the fields $k_1=\F_{p}$ and $k_2= \F_{p^2}$ of all the sub-extensions of $F$ defined over $\F_{p^4}$. We give explicit examples with small prime numbers $p$.