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, Volume 9 (1997) no. 1, pp. 201-219.

Let K be a finite extension of 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(KΓ;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 -1 (mod p n ) for all i. We give three consequences of this. Firstly, if 𝔄(KΓ;S) is a Hopf order and S is 𝔄(KΓ;S)-Galois then t i -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 p , then S is not free over (𝔄(KΓ;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 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(KΓ;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 -1 (mod p n ) pour tout i. Nous déduisons de cela trois conséquences. Premièrement, si 𝔄(KΓ;S) est un ordre de Hopf et si S/R est une 𝔄(KΓ;S)-extension galoisienne, où R est l’anneau de valuation de K, alors t i -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 p , alors S n’est pas libre sur 𝔄(KΓ;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
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Byott, Nigel P. 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, Volume 9 (1997) no. 1, pp. 201-219. http://www.numdam.org/item/JTNB_1997__9_1_201_0/

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