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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.
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
Full Text: DOI
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