On von Staudt for Bernoulli–Carlitz numbers

In 1935, Carlitz introduced analogues of Bernoulli numbers for Fq[t]. These are now called Bernoulli–Carlitz numbers Bm. He proved a von Staudt type theorem, with a much more subtle statement than the classical one, describing their denominators completely. As an analog of the important relative Bm/m of the usual Bernoulli number Bm, Thakur considered an analog Bm(m−1)!C/m!C, where m!C is the Carlitz factorial. He described their denominator fully except when q=2 and m has a particular form. The purpose of this paper is to completely describe this last remaining situation. Also, we shall see that a group of symmetries recently discovered by Goss may be realized as symmetries of our results.