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Arithmetic actions on cyclotomic function fields. (English) Zbl 07241688
Let $${\mathbb F}_q$$ be the finite field of $$q$$ elements and $$d$$ a positive integer. Let $${\mathbb F}_{q^d}(T)$$ be the extension of constants of degree $$d$$ of $${\mathbb F}_q(T)$$. Let $$C_{q^d}(T)(u):=Tu+u^{q^d}$$, for $$u\in\overline{{\mathbb F}_{q^d}(T)}$$, be the Carlitz action. Denote by $$C_{q^d}[M]$$ the $$M$$-torsion for $$M\in{\mathbb F}_{q^d}[T]$$. Let $$K_{q^d,M}={\mathbb F}_{q^d}(T)(C_{q^d}[M])$$ be the $$M$$-cyclotomic function field over $${\mathbb F}_{q^d}$$. In what follows, it is considered $$M\in{\mathbb F}_q[T]$$ such that $$M$$ splits completely over $${\mathbb F}_{q^d}[T]$$. The first main result is that $$K_{q,M}\subseteq K_{q^d,M}$$. The main tool in the proof is the use of Bauer’s theorem that establishes that for any global field $$K$$ and two finite Galois extensions $$L_1$$ and $$L_2$$ of $$K$$, we have $$L_2\subseteq L_1$$ if and only if $$S(L_1/K)\subseteq S(L_2/K)$$ where $$S(L_i/K)$$ denote the set of places of $$K$$ which split completely in $$L_i$$.
Next, the authors study $$H_{q^d,M}:={\mathrm{Gal}}(K_{q^d,M}/{\mathbb F}_{q^d}K_{q,M})$$ in Section 2. The group $${\mathrm{Gal}}(K_{q^d,M}/{\mathbb F}_{q^d}(T))$$ is naturally a $${\mathrm{Gal}}({\mathbb F}_{q^d}/{\mathbb F}_q)=\langle \sigma\rangle$$-module and it is shown that $$H_{q^d,M}=(\sigma-1) {\mathrm{Gal}}(K_{q^d,M}/{\mathbb F}_{q^d}(T))$$. In Section 3, the authors give an explicit generation of the tame part of $$H_{q^d,M}$$ and in Section 4 they describe the wild component of $$H_{q^d,M}$$, including the higher ramification groups and the different.
Section 6 is devoted to describe the Galois module structure of the differentials of $$K_{q,M}$$. The space of holomorphic differentials $$H^0(X,\Omega_X)$$ was computed by the second author in [K. A. Ward, Finite Fields Appl. 44, 34–55 (2017; Zbl 1359.14034)] for a curve $$X$$ corresponding to $$K_{q^d,M}$$ for $$d$$ big enough so that $$M$$ splits completely in $${\mathbb F}_{q^d}$$. To compute $$H^0(Y,\Omega_Y)$$ for the curve $$Y$$ corresponding to $$K_{q,M}$$, where $$M$$ does not split in $${\mathbb F}_q(T)$$, it is considered the Galois cover $$X\longrightarrow Y$$ with Galois group $$H=H_{q,M}$$ and reduced holomorphic differentials of $$X$$ to holomorphic differentials of $$Y$$. It is shown that $$H^0(Y,\Omega_Y)\subseteq L_Y(\Omega(D))=H^0(X,\Omega_X)^H$$, where $$L_Y(\Omega(D))$$ is a space of differentials with poles on a certain divisor $$D$$. The second author showed [loc. cit.] that holomorphic differentials on certain curves $$X$$ are given by generators $$\lambda_{i,k}$$ of the Carlitz torsion modules $$C_{q^d}[P_i^k]$$, where $$M=\prod_{i=1}^r P_i^{n_i}$$. The last main result is that the ring of invariants for the algebra of Carlitz generators is not polynomial.
##### MSC:
 11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.) 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11R58 Arithmetic theory of algebraic function fields 13N05 Modules of differentials 14H37 Automorphisms of curves 14H55 Riemann surfaces; Weierstrass points; gap sequences
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