Lubin-Tate formal groups and module structure over Hopf orders
Journal de théorie des nombres de Bordeaux, Volume 11 (1999) no. 2, pp. 269-305.

Over the last years Hopf orders have played an important role in the study of integral module structures arising in arithmetic geometry in various situations. We axiomatize these situations and discuss the properties of the (integral) Hopf algebra structures which are of interest in this general setting. In particular, we emphasize the role of resolvents for explicit computations. As an illustration we apply our results to determine the Hopf module structure of the ring of integers in relative Lubin-Tate extensions.

Ces dernières années les ordres de Hopf ont joué dans des situations diverses un rôle important dans l'étude de la structure des module galoisiens en géométrie arithmétique. Nous introduisons ici un cadre qui rend compte des situations précédentes, et nous étudions les propriétés des algèbres de Hopf dans ce contexte général. Nous insistons en particulier sur le rôle des résolvantes dans les calculs explicites. Nous illustrons cette étude en appliquant nos résultats à la détermination de la structure de module de Hopf de l'anneau des entiers d'une extension de Lubin-Tate relative.

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     title = {Lubin-Tate formal groups and module structure over {Hopf} orders},
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     publisher = {Universit\'e Bordeaux I},
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Bley, Werner; Boltje, Robert. Lubin-Tate formal groups and module structure over Hopf orders. Journal de théorie des nombres de Bordeaux, Volume 11 (1999) no. 2, pp. 269-305. http://www.numdam.org/item/JTNB_1999__11_2_269_0/

[A] A. Agboola, Torsion points on elliptic curves and galois module structure. Invent. Math. 123 (1996), 105-122. | MR | Zbl

[B] W. Bley, Elliptic curves and module structure over Hopf orders and The conjecture of Chinburg-Stark for abelian extensions of a quadratic imaginary field. Habilitation Thesis Universität Augsburg, Report des Instituts für Mathematik der Universität Augsburg No. 396, 1998.

[BT] N. Byott, M.J. Taylor, Hopf orders and Galois module structure. In: Group rings and class groups, R. W. Roggenkamp, M. J. Taylor (eds.) Birkhäuser, Basel Boston, 1992. | MR | Zbl

[By] N. Byott, Associated orders of certain extensions arising fom Lubin-Tate formal groups. J. Théor. Nombres Bordeaux 9 (1997),449-462. | Numdam | MR | Zbl

[CT] Ph. Cassou-Noguès, M.J. Taylor, Elliptic functions and rings of integers. Prog. in Math. 66, Basel-Stuttgart-Boston, 1987. | MR | Zbl

[Ch] Sh.-P. Chan, Relative Lubin-Tate formal groups and Galois module structure. Manuscripta Math. 39 (1992), 109-113. | MR | Zbl

[CL] Sh.-P. Chan, C.-H. Lim, The associated orders of rings of integers in Lubin-Tate division fields over the p-adic number field. Illinois J. Math. 39 (1995), 30-38. | MR | Zbl

[CS] S.U. Chase, M.E. Sweedler, Hopf algebras and Galois theory. Springer Lecture Notes in Mathematics 97, Springer-Verlag, 1969. | MR | Zbl

[CH] L.N. Childs, S. Hurley, Tameness and local normal bases for objects of finite Hopf algebras. Trans. Amer. Math. Soc. 298 (1986), 763-778. | MR | Zbl

[dS] E. Deshalit, Iwasawa Theory of Elliptic Curves with Complex Multiplication. Perspectives in Math. Vol. 3, Academic Press, 1987. | MR | Zbl

[R] I. Reiner, Maximal orders., Academic Press, 1975. | MR | Zbl

[S] R. Schertz, Galoismodulstruktur und Elliptische Funktionen. J. Number Theory 39 (1991), 285-326. | MR | Zbl

[ST] A. Srivastav, M.J. Taylor, Elliptic curves with complez multiplication and Galois module structure. Invent. Math. 99 (1990), 165-184. | MR | Zbl

[T1] M.J. Taylor, Hopf Structure and the Kummer Theory of Formal Groups. J. Reine Angew. Math. 375/376 (1987), 1-11. | MR | Zbl

[T2] M.J. Taylor, Mordell-Weil Groups and the Galois Module Structure of Rings of Integers. Illinois J. Math. 32 (1988), 428-452. | MR | Zbl