On the orthogonal of cyclotomic units in positive characteristic (Q1970608)

This is a very interesting and powerful piece of mathematics. Let \(A:= \mathbb{F}_q[T]\) where \(\mathbb{F}_q\) is the finite field of \(q=p^m\) elements. Let \(P\in A\) be a monic prime of degree \(d\). In work by the author [J. Number Theory 78, 228-252 (1999; Zbl 0937.11020)] and \textit{F. Schultheis} [J. Number Theory 52, 119-124 (1995; Zbl 0839.11058)] a local symbol attached to the Carlitz module was studied. Let \(F\) be the maximal ``real'' (i.e., totally split at \(\infty\)) subfield of the cyclotomic function field associated to \(F\). In the paper being reviewed, this local symbol is used to define the ``orthogonal space'' \(\text{Cyc}_F^\perp\) associated to the cyclotomic units of \(F\). Set \(\ell:= [F:\mathbb{F}_q(T)]\) and let \(r_\ell(P)\) be the number of Bernoulli-Carlitz numbers \(B(k)\) such that \(0< k< q^d-1\) is divisible by \((q^d-1)/\ell\) and \(B(k)\) is not divisible by \(P\). Let \(E_F^\perp\) be the orthogonal of the full group of units of the ring of \(A\)-integers of \(F\). Then, inspired by work of \textit{S. V. Vostokov} [Transl., Ser. 2, Am. Math. Soc. 166, 149-156 (1995; Zbl 0894.11044)], the author shows that the space \(\text{Cyc}_F^\perp/ E_F^\perp\) is an \(A/P\)-vector space of dimension \(\leq\ell-1- r_\ell(P)\). In the last section the author establishes an analog of the well-known result of Ankeny-Artin-Chowla when \(q\) is odd. In particular, the author establishes the beautiful result that the ideal class number of the ring \(\mathbb{F}_q [T,\sqrt{P}]\) is divisible by \(p\) if and only if \(P\) divides \(B((q^d- 1)/2)\).