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Application of the Hodge-Tate dual of a Lubin-Tate group, Bruhat-Tits building of the linear group and ramification filtrations. (Application de Hodge-Tate duale d’un groupe de Lubin-Tate, immeuble de Bruhat-Tits du groupe linéaire et filtrations de ramification.) (French) Zbl 1136.14013
Let \(F\) be a finite extension of \(\mathbb Q_p\). Consider the following two moduli spaces of formal groups. Both are rigid analytic spaces. The Drinfeld moduli space \(\Omega\) parametrizes formal groups of dimension \(n\) and height \(n^2\) with an action of the integers of the division algebra \(D\) over \(F\) with invariant \(1/n\), and its \(\mathbb C_p\)-valued points are the complement of all rational hyperplanes in \(\mathbb P^{n-1} (\mathbb C_p)\). The Lubin-Tate space parametrizes formal groups of dimension \(1\) and height \(n\), and it is an open \(p\)-adic ball \({^{\circ}{\mathbb B}}^{n-1}\) in the sense of Berkovich. By adding level structures to the moduli problems in question, both moduli problems can be extended to infinite towers.
G. Faltings [in: Algebraic number theory and algebraic geometry. Papers dedicated to A. N. Parshin on the occasion of his sixtieth birthday. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 300, 115–129 (2002; Zbl 1062.14059)] has proved that, in a suitable sence, the limit objects of both towers are isomorphic in an equivariant way for the action of \(\mathrm{GL}_n(F)\times D^\times\). In particular, the associated Berkovich spaces are homeomorphic. In [L. Fargues, A. Genestier and V. Lafforgue, “The isomorphism between Lubin-Tate and Drinfeld towers.” Progr. Math. 262 (Base)l: Birkhäuser. 1–325 (2008; Zbl 1136.14001)], the author has worked out a detailed proof of this theorem.
After passing to the quotient by \(\mathrm{GL}_n(\mathcal O_F)\times \mathcal O_D^\times\), one gets a map \(\mathcal O_D^\times \backslash | {^{\circ}{\mathbb B}}^{n-1} | \rightarrow \mathrm{GL}_n(\mathcal O_F)\backslash | \Omega |\). In the paper at hand, this map is investigated at the level of the “skeletons” of the two sides. The skeleton in the Drinfeld case is the quotient of the Bruhat-Tits building of \(\mathrm{PGL}_n\) by the action of \(\mathrm{GL}_n(\mathcal O_F)\). In the Lubin-Tate case, the skeleton is the space \((0,\infty]^{n-1}\), with the simplicial structure induced from “Newton polygons”. In both cases, there are natural retractions from the space in question to its skeleton, and it is proved that the isomorphism above induces a map between the skeletons which gives an isomorphism after a small correction.
Along the way, the ramification filtrations and the Hodge-Tate map of a one-dimensional formal group are studied. The simplicial structure one obtains on the Lubin-Tate space in this way is very useful, and in the paper applications to the study of canonical subgroups, the description of Hecke orbits, fundamental domains for Hecke correspondences, and the period mapping are given. In the final section, the results are transferred to the Iwahori case.

14G35 Modular and Shimura varieties
14L05 Formal groups, \(p\)-divisible groups
Full Text: DOI
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