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

Byott, Nigel P.

J. Théor. Nombres Bordx. 9, No. 2, 449-462 (1997).

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)

MSC:

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

Keywords:

associated orders; Galois modules; ramification; Lubin-Tate extensions; divisibility properties of binomial coefficients

Citations:

Zbl 0889.11040

References:

[1]	Byott, N.P., Some self-dual rings of integers not free over their associated orders, Math. Proc. Camb. Phil. Soc.110 (1991), 5-10; Corrigendum, 116 (1994), 569. · Zbl 0737.11037
[2]	Byott, N.P., Galois structure of ideals in wildly ramified abelian p-extensions of a p-adic field, and some applications, J. de Théorie des Nombres de Bordeaux9 (1997), 201-219. · Zbl 0889.11040
[3]	Chan, S.-P., Galois module structure of non-Kummer extensions, Preprint, National University of Singapore (1995).
[4]	Chan, S.-P. and Lim, C.-H., The associated orders of rings of integers in Lubin- Tate division fields over the p-adic number field, Ill. J. Math.39 (1995), 30-38. · Zbl 0816.11061
[5]	Ribenboim, P., The Book of Prime Number Records, 2nd edition, Springer, 1989. · Zbl 0642.10001
[6]	Serre, J.-P., Local Class Field Theory, in Algebraic Number Theory (J.W.S. Cassels and A. Fröhlich, eds.), Academic Press, 1967. · Zbl 1492.11158
[7]	Taylor, M.J., Formal groups and the Galois module structure of local rings of integers, J. reine angew. Math.358 (1985), 97-103. · Zbl 0582.12008