Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications

Byott, Nigel P.

Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 201-219

Let $K$ be a finite extension of ${\mathbb{Q}}_p$ with ramification index $e$, and let $L/K$ be a finite abelian $p$-extension with Galois group $\Gamma$ and ramification index $p^n$. We give a criterion in terms of the ramification numbers $t_i$ for a fractional ideal ${𝔓}^h$ of the valuation ring $S$ of $L$ not to be free over its associated order $𝔄(K\Gamma ;{𝔓}^h)$. In particular, if $t_n-[t_n/p]<p^{n-1}e$ then the inverse different can be free over its associated order only when $t_i\equiv -1$ (mod $p^n$) for all $i$. We give three consequences of this. Firstly, if $𝔄(K\Gamma ;S)$ is a Hopf order and $S$ is $𝔄(K\Gamma ;S)$-Galois then $t_i\equiv -1$ (mod $p^n$) for all $i$. Secondly, if $K=k_{r}L=k_{m+r}$ are Lubin-Tate division fields, with $m>r$ and $k\ne {\mathbb{Q}}_p$, then $S$ is not free over ($𝔄(K\Gamma ;S)$. Thirdly, these extensions $k_{m+r}/k_r$ admit two Hopf Galois structures exhibiting different behaviour at integral level.

Soit $K$ une extension finie de ${\mathbb{Q}}_p$ d’indice de ramification $e$, et soit $L/K$ une $p$-extension abélienne finie de groupe de Galois $\Gamma$ et d’indice de ramification $p^n$. Nous donnons un critère en termes des nombres de ramification $t_i$ permettant de décider lorsqu’un idéal fractionnaire ${𝔓}^h$ de l’anneau de valuation $S$ de $L$ peut être libre sur son ordre associé $𝔄(K\Gamma ;{𝔓}^h)$. En particulier, si $t_n-[t_n/p]<p^{n-1}e$, la codifférente ne peut être libre sur son ordre associé que si $t_i\equiv -1$ (mod $p^n$) pour tout $i$. Nous déduisons de cela trois conséquences. Premièrement, si $𝔄(K\Gamma ;S)$ est un ordre de Hopf et si $S/R$ est une $𝔄(K\Gamma ;S)$-extension galoisienne, où $R$ est l’anneau de valuation de $K$, alors $t_i\equiv -1$ (mod $p^n$) pour tout $i$. Deuxièmement, si $K=k_r$ et $L=k_{m+r}$ sont des corps de points de division d’un groupe de Lubin-Tate, avec $m>r$ et $k\ne {\mathbb{Q}}_p$, alors $S$ n’est pas libre sur $𝔄(K\Gamma ;S)$. Troisièmement, ces extensions $k_{m+r}/k_r$ possèdent deux structures galoisiennes de Hopf différentes, mettant en évidence des comportements différents au niveau des entiers.

Classification : 11S23, 11R33, 11S31, 16W30
Keywords: Galois module structure, associated order, Hopf order, Lubin-Tate formal group

[Be] A.-M. Bergé, Arithmétique d'une extension galoisienne à groupe d'inertie cyclique, Ann. Inst. Fourier, Grenoble 28 (1978), 17-44. | Numdam | Zbl | MR | EuDML

[B-F] F. Bertrandias and M.-J. Ferton, Sur l'anneau des entiers d'une extension cyclique de degré premier d'un corps local, C. R. Acad. Sc. Paris 274 (1972), A1330-A1333. | Zbl | MR

[Bl-Bu] W. Bley and D. Burns, Über arithmetische assoziierte Ordnungen, J. Number Theory 58 (1996), 361-387. | Zbl | MR

[Bu1] D. Burns, Factorisability and wildly ramified Galois extensions, Ann. Inst. Fourier, Grenoble 41 (1991), 393-430. | Numdam | Zbl | MR | EuDML

[Bu2] D. Burns, On the equivariant structure of ideals in Galois extensions of fields, Preprint, King's College London (1996).

[By1] N.P. Byott, 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. | MR | Zbl

[By2] N. Byott, On Galois isomorphisms between ideals in extensions of local fields, Manuscripta Math. 73 (1991), 289-311. | Zbl | MR

[By3] N.P. Byott, Tame and Galois extensions with respect to Hopf orders, Math. Z. 220 (1995), 495-522. | Zbl | MR

[By4] N.P. Byott, Uniqueness of Hopf Galois structure for separable field extensions, Comm. Alg. 24 (1996), 3217-3228; Corrigendum 24 (1996), 3705. | Zbl | MR

[By5] N.P. Byott, Associated orders of certain extensions arising from Lubin- Tate formal groups, to appear in J. de Théorie des Nombres de Bordeaux. | Zbl | MR | Numdam

[By-L] N.P. Byott and G. Lettl, Relative Galois module structure of integers in abelian fields, J. de Théorie des Nombres de Bordeaux 8 (1996), 125-141. | Zbl | MR | Numdam

[C-L] S.-P. Chan and C.-H. Lim, 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 | MR

[C] L.N. Childs, Taming wild extensions with Hopf algebras, Trans. Am. Math. Soc. 304 (1987), 111-140. | Zbl | MR

[C-M] L.N. Childs and D.J. Moss, Hopf algebras and local Galois module theory, in Advances in Hopf Algebras, Lect. Notes Pure and Appl. Math. Series, Vol. 158 (J. Bergen and S. Montgomery, eds.), Dekker, 1994, pp. 1-14. | Zbl | MR

[E] G.G. Elder, Galois module structure of ideals in wildly ramified cyclic extensions of degree p2, Ann. Inst. Fourier, Grenoble 45 (1995), 625-647. | Zbl | MR | Numdam

[E-M] G.G. Elder and M.L. Madan, Galois module structure of the integers in wildly ramified cyclic extensions, J. Number Theory 47 (1994), 138-174. | Zbl | MR

[F] M.-J. Ferton, Sur les idéaux d'une extension cyclique de degré premier d'un corps local, C. R. Acad. Paris 276 (1973), A1483-A1486. | Zbl | MR

[G] C. Greither, Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring, Math. Z. 210 (1992), 37-67. | Zbl | MR

[G-P] C. Greither and B. Pareigis, Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), 239-258. | Zbl | MR

[L] H.-W. Leopoldt, Uber die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. reine u. angew. Math. 201 (1959), 119-149. | Zbl | MR

[RC-VS-M] M. Rzedowski Calderón, G.D. Villa Salvador and M.L. Madan, Galois module structure of rings of integers, Math. Z. 204 (1990), 401-424. | Zbl | MR

[S1] J.-P. Serre, Local Class Field Theory, in Algebraic Number Theory (J.W.S. Cassels and A. Fröhlich, eds.), Academic Press, 1967. | MR

[S2] J.-P. Serre, Local fields (Graduate Texts in Mathematics, Vol. 67), Springer, 1979. | Zbl | MR

[T1] M.J. Taylor, Formal groups and the Galois module structure of local rings of integers, J. reine angew. Math. 358 (1985), 97-103. | Zbl | MR

[T2] M.J. Taylor, Hopf structure and the Kummer theory of formal groups, J. reine angew. Math. 375/376 (1987), 1-11. | Zbl | MR

[U] S. Ullom, Integral normal bases in Galois extensions of local fields, Nagoya Math. J. 39 (1970), 141-148. | Zbl | MR

[V1] S.V. Vostokov, Ideals of an abelian p-extension of an irregular local field as Galois modules, Zap. Nauchn. Sem. Lening. Otdel. Math. Inst. Steklov. (LOMI) 46 (1974), 14-35; English transl. in J. Soviet Math. 9 (1978), 299-317. | Zbl | MR

[V2] S.V. Vostokov, Ideals of an abelian p-extension of a local field as Galois module, Zap. Nauchn. Sem. Lening. Otdel. Math. Inst. Steklov. (LOMI) 57 (1976), 64-84; English transl. in J. Soviet Math. 11 (1979), 567-584. | Zbl | MR

[W] W.C. Waterhouse, Normal basis implies Galois for coconnected Hopf algebras, Preprint, Pennsylvania State University (1992).