Associated orders of certain extensions arising from Lubin-Tate formal groups. (English) Zbl 0902.11052

In a previous paper [N. P. Byott, J. Théor. Nombres Bordx. 9, No. 1, 201-219 (1997; Zbl 0889.11040)], the author proved that in non-Kummer Lubin-Tate extensions \(L/K\), the ring of integers in \(L\) is almost never free over its associated order in \(K[G(L/K)]\). His clever proof did not require an explicit description of the associated order. The present paper can be considered as a complement: here the author considers the minimal non-Kummer example (\(L=\) field of \(\pi^3\)-torsion points, \(K=\) field of \(\pi\)-torsion points), and he gives a complete and utterly explicit calculation of the associated order, under a hypothesis concerning the absolute ramification degree. To do this, he starts from some sums in the group ring \(K[G(L/K)]\) which have also been considered by Taylor; one major point is to find the maximal power of \(\pi_K\) dividing one of these sums in the associated order. This then becomes very technical: among other things, some divisibility properties of binomial coefficients have to be studied. This is a highly specialized paper, but again, a well-written one.
Reviewer: C.Greither (Laval)


11S23 Integral representations
11S31 Class field theory; \(p\)-adic formal groups
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers


Zbl 0889.11040
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