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Carlitz-Kummer function fields. (English) Zbl 0712.11067

Let \(L\) be a field and let \(m\) be a positive integer prime to the characteristic of \(L\). Assume that \(L\) contains a primitive \(m\)-th root of unity. One then has the basic Kummer duality which describes abelian extensions of \(L\) of exponent \(m\) in terms of subgroups of \(L^*\) which contain \(L^{*m}\). This description is accomplished by taking \(m\)-th roots, etc.
Now let \(L\) have characteristic \(p>0\). Let \(\wp(z)\) be the operator defined on the additive group of \(L\) by \(\wp(z)=z^p-z\). One then has the additive Kummer theory which describes abelian extensions of \(L\) of exponent \(p\) in terms of additive subgroups of \(L\) which contain \(\wp(L)\). The analogy with the multiplicative case is exact.
Recall that in characteristic-\(p\) taking \(p\)-th roots never leads to separable non-trivial extensions. Thus Kummer theory for extensions of exponent \(p\) is a strictly additive phenomenon (i.e., one must use operators like \(\wp(z)\) etc.). On the other hand, there is another natural way to construct abelian extensions which arises from the theory of Drinfeld modules. The paper under review is concerned with such a study. The author restricts himself to the most basic case where the Drinfeld module is the Carlitz module, \(C\), for \(A=F_q[T]\) defined by \(C_T(z)=Tz+z^q\).
The idea is as follows: Let \(M\) be a monic polynomial in \(F_q[T]\) and let \(\Lambda_M\) be the roots of \(C_M(z)=0\) in some fixed algebraic closure of \(L\). It is basic to theory that \(\Lambda_M\) inherits a natural \(A\)-module structure via \(C\). Now let \(K\) be any finite extension of \(L\) which contains \(\Lambda_M\) and let \(\alpha\) be any element of \(K\) not in \(C_M(K)\). The author then calls the splitting field of \(C_M(z)-\alpha\) a Carlitz-Kummer extension of \(K\). From the definition it is easy to see that this extension is abelian of \(p\)-power order and that the Galois group is isomorphic to a subgroup of \(\Lambda_M\). The ramification properties of these extensions are then studied and much useful information is presented.
The author mentions that, with the collaboration of D. J. Lewis, explicit reciprocity laws for “\(M\)-th power” residue symbols have been computed. No details are presented here, however.
Two questions arise from the analogy with classical theory:
1. Is there some sort of intrinsic characterization of the abelian extensions obtained by the Carlitz-Kummer construction?
2. Along the same lines, how can one put the A-module structure back into the theory? (E.g., as we saw above the Galois groups correspond to subgroups and not submodules.)

MSC:

11R58 Arithmetic theory of algebraic function fields
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.)
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References:

[1] Carlitz, L., A class of polynomials, Trans. Amer. Math. Soc., 43, 167-182 (1938) · JFM 64.0093.01
[2] Galovich, S.; Rosen, M., The class number of cyclotomic function fields, J. Number Theory, 13, 363-375 (1981) · Zbl 0473.12014
[3] Hayes, D., Explicit class field theory for rational function fields, Trans. Amer. Math. Soc., 189, 77-91 (1974) · Zbl 0292.12018
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