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

Nigel P. Byott

doi:10.5802/jtnb.196

Galois structure of ideals in wildly ramified abelian

p

-extensions of a

p

-adic field, and some applications

Nigel P. Byott

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

Résumé
Abstract

Soit $K$ une extension finie de $ℚ_{p}$ d’indice de ramification $e$ , et soit $L / K$ une $p$ -extension abélienne finie de groupe de Galois $Γ$ 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 $𝔄 (K Γ; 𝔓^{h})$ . 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 $𝔄 (K Γ; S)$ est un ordre de Hopf et si $S / R$ est une $𝔄 (K Γ; S)$ -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 $k \neq ℚ_{p}$ , alors $S$ n’est pas libre sur $𝔄 (K Γ; S)$ . 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.

Let $K$ be a finite extension of $ℚ_{p}$ with ramification index $e$ , and let $L / K$ be a finite abelian $p$ -extension with Galois group $Γ$ 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 $𝔄 (K Γ; 𝔓^{h})$ . 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 $𝔄 (K Γ; S)$ is a Hopf order and $S$ is $𝔄 (K Γ; S)$ -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 $k \neq ℚ_{p}$ , then $S$ is not free over ( $𝔄 (K Γ; S)$ . Thirdly, these extensions $k_{m + r} / k_{r}$ admit two Hopf Galois structures exhibiting different behaviour at integral level.

MR 1469668 | Zbl 0889.11040

DOI : https://doi.org/10.5802/jtnb.196
Classification : 11S23, 11R33, 11S31, 16W30
Mots clés : Galois module structure, associated order, Hopf order, Lubin-Tate formal group

@article{JTNB_1997__9_1_201_0,
     author = {Byott, Nigel P.},
     title = {Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {201--219},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {1},
     year = {1997},
     doi = {10.5802/jtnb.196},
     zbl = {0889.11040},
     mrnumber = {1469668},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.196/}
}

Nigel P. Byott. Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications. Journal de Théorie des Nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 201-219. doi : 10.5802/jtnb.196. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.196/

Bibliographie

[Be] A.-M. Bergé, Arithmétique d'une extension galoisienne à groupe d'inertie cyclique, Ann. Inst. Fourier, Grenoble 28 (1978), 17-44. | EuDML 74380 | Numdam | MR 513880 | Zbl 0377.12009

[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. | MR 296047 | Zbl 0235.12007

[Bl-Bu] W. Bley and D. Burns, Über arithmetische assoziierte Ordnungen, J. Number Theory 58 (1996), 361-387. | MR 1393621 | Zbl 0857.11063

[Bu1] D. Burns, Factorisability and wildly ramified Galois extensions, Ann. Inst. Fourier, Grenoble 41 (1991), 393-430. | EuDML 74923 | Numdam | MR 1137291 | Zbl 0727.11048

[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 1104596 | Zbl 0836.11039

[By2] N. Byott, On Galois isomorphisms between ideals in extensions of local fields, Manuscripta Math. 73 (1991), 289-311. | MR 1132141 | Zbl 0771.11047

[By3] N.P. Byott, Tame and Galois extensions with respect to Hopf orders, Math. Z. 220 (1995), 495-522. | MR 1363852 | Zbl 0841.16021

[By4] N.P. Byott, Uniqueness of Hopf Galois structure for separable field extensions, Comm. Alg. 24 (1996), 3217-3228; Corrigendum 24 (1996), 3705. | MR 1402555 | Zbl 0878.12001

[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. | Numdam | MR 1617408 | Zbl 0902.11052

[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. | Numdam | MR 1399950 | Zbl 0859.11059

[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. | MR 1299647 | Zbl 0816.11061

[C] L.N. Childs, Taming wild extensions with Hopf algebras, Trans. Am. Math. Soc. 304 (1987), 111-140. | MR 906809 | Zbl 0632.12013

[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. | MR 1289419 | Zbl 0826.16035

[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. | Numdam | MR 1340947 | Zbl 0820.11070

[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. | MR 1275759 | Zbl 0801.11046

[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. | MR 332733 | Zbl 0268.12006

[G] C. Greither, Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring, Math. Z. 210 (1992), 37-67. | MR 1161169 | Zbl 0737.11038

[G-P] C. Greither and B. Pareigis, Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), 239-258. | MR 878476 | Zbl 0615.12026

[L] H.-W. Leopoldt, Uber die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. reine u. angew. Math. 201 (1959), 119-149. | MR 108479 | Zbl 0098.03403

[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. | MR 1107472 | Zbl 0682.12003

[S1] J.-P. Serre, Local Class Field Theory, in Algebraic Number Theory (J.W.S. Cassels and A. Fröhlich, eds.), Academic Press, 1967. | MR 220701

[S2] J.-P. Serre, Local fields (Graduate Texts in Mathematics, Vol. 67), Springer, 1979. | MR 554237 | Zbl 0423.12016

[T1] M.J. Taylor, Formal groups and the Galois module structure of local rings of integers, J. reine angew. Math. 358 (1985), 97-103. | MR 797677 | Zbl 0582.12008

[T2] M.J. Taylor, Hopf structure and the Kummer theory of formal groups, J. reine angew. Math. 375/376 (1987), 1-11. | MR 882287 | Zbl 0609.12015

[U] S. Ullom, Integral normal bases in Galois extensions of local fields, Nagoya Math. J. 39 (1970), 141-148. | MR 263790 | Zbl 0199.08401

[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. | MR 371865 | Zbl 0396.12016

[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. | MR 453708 | Zbl 0403.12017

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