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Units and class groups in cyclotomic function fields. (English) Zbl 0483.12003

Let \(p\) be an odd prime. Let \(w\) denote a primitive \(p\)-th root of 1. In the mid 19th century, Kummer proved that the class number of \(\mathbb Z[x + w^{-1}]\) (the ring of integers of the maximal real subfield of \(\mathbb Z[w])\) equals the index in \(\mathbb Z[w]^*\) (the unit group of \(\mathbb Z[w])\) of the subgroup of circular units. The latter is the subgroup of \(\mathbb Z[w]^*\) generated by \((1 - w^n)/( 1 - w)\) for \(1\le n\le p-1\). In 1978 this theorem was generalized by Sinnott to the case that \(w\) is a primitive \(m\)-th root of 1 where \(m\) is an arbitrary positive integer.
Let \(\mathbb F\) be a field with \(q\) elements and let \(k = \mathbb F(T)\) be a rational function field in one variable over \(\mathbb F\). For each monic polynomial \(M\in \mathbb F[T]\), Carlitz associated an abelian extension \(k(M)\) of \(k\). The field \(k(M)\) is analogous in many ways to the cyclotomic extension \(\mathbb Q(w)\) of \(\mathbb Q\).
In this paper the authors prove an analogue of Sinnott’s theorem for the “cyclotomic function field” \(k(M)\). First, they define the “maximal real subfield” \(F(M)\) of \(k(M)\). They establish basic arithmetic properties and analytic class number formulas for the fields \(k(M)\) and \(F(M)\). Let \(\mathfrak O\) denote the integral closure of \(\mathbb F[T]\) in \(k(M)\). The authors define a subgroup \(C\) of \(\mathfrak O^*\) (the units of \(\mathfrak O)\) in analogy with the circular units in cyclotomic number fields. Let \(h\) denote the class number of the integral closure of \(\mathbb F[T]\) in \(f(M)\) and let \(g\) denote the number of monic prime factors of the polynomial \(M\). The principal result is
\[ [\mathfrak O^* : C] = h\cdot (q - 1)^a \]
where \(a = 0\) if \(g = 1\) and \(a = 2^{g-2}+1- g\) if \(g>1\). The proof of this formula involves the study of distributions on \(k\) and an analysis of certain related cohomology groups.

MSC:

11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.)
11R18 Cyclotomic extensions
11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
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