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Bruhat-Tits Theory from Berkovich's Point of View. Analytic Filtrations

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Bruhat-Tits Theory from Berkovich's Point of View. Analytic Filtrations Bruhat-Tits theory from Berkovich’s point of view. Analytic filtrations Arnaud Mayeux To cite this version: Arnaud Mayeux. Bruhat-Tits theory from Berkovich’s point of view. Analytic filtrations. 2020. hal-02566872v1 HAL Id: hal-02566872 https://hal.archives-ouvertes.fr/hal-02566872v1 Preprint submitted on 7 May 2020 (v1), last revised 16 Nov 2020 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Bruhat-Tits theory from Berkovich's point of view. Analytic ltrations Arnaud Mayeux Let k be a p-adic eld. We dene ltrations by k-anoid groups, in the Berkovich analytication Gan of a connected reductive group G=k, related to Moy-Prasad ltrations. They are parametrized by a cone, whose basis is the Bruhat-Tits building and whose vertex is the neutral element, in Gan via the notions of Shilov boundary and holomorphically convex envelope. Contents 1 Schematic congruence subgroups 4 2 Berkovich k-analytic spaces 7 2.1 Analytication of k-schemes and k◦-schemes . .7 2.2 Analytic groups and descent . 10 3 Bruhat-Tits buildings and Moy-Prasad ltrations 11 3.1 Rational points . 11 3.2 Demazure group schemes . 12 3.3 Moy-Prasad ltrations . 12 4 Denitions and rst properties of analytic ltrations 12 4.1 Notions of potentially Demazure objects . 12 4.2 Filtrations of rational potentially Demazure k-anoid groups 14 4.2.1 The split rational case . 14 4.2.2 The general case . 18 4.3 Filtrations of Lie algebra . 22 5 Filtrations associated to points in the Bruhat-Tits building 22 5.1 Denitions and properties of Gx;r ............... 22 5.2 Denitions and properties of the map θ ............ 24 5.3 A cone . 28 5.4 Comparison with Moy-Prasad ltrations in the tame case . 29 5.5 Filtrations of the Lie algebra . 30 5.6 Examples and pictures . 30 5.6.1 The split torus of rank one . 30 5.6.2 A computation of Gx;0 in the case of a wild torus of norm one elements in a quadratic extension . 33 Appendix: On notions of rational points 36 1 Acknowledgments I thank Anne-Marie Aubert, Marco Maculan and Antoine Ducros for help and comments. This research work was done during author's thesis at IMJ- PRG. Introduction Let G be a connected reductive group over k. Berkovich [1, chapter 5] (split case) and Rémy-Thuillier-Werner [14] (general case) showed that there is a canonical embedding of the Bruhat-Tits building of G into the Berkovich analytication Gan of G. This embedding and related ideas form what is called Berkovich's point of view on Bruhat-Tits buildings . In this article, we observe that Berkovich's point of view allows to dene and parametrize natural k-analytic ltrations related to Moy-Prasad ltrations [11] [12]. Let x be a point in the reduced Bruhat-Tits building of G over k. The group G(k) acts on the reduced and enlarged buildings. This action is compatible with the canonical projection from the enlarged building to the reduced one. Let us consider the stabilizer stabg (x) of a preimage of x in the enlarged building, this is a compact open subgroup of G(k) independent of the preimage. The idea of Berkovich's point of view is to construct a k-anoid group Gx which realizes stabg (x). The space Gx is equipped with an ordered relation and has a maximal point: its Shilov Boundary denoted θ(x). The space Gx can be recovered from θ(x) taking the holomorphically convex envelope. The preceding constructions and results for the general case are done in [14]. Now our ltrations are certains k-anoid groups Gx;r contained in Gx (we will assume that x and r are rational) satisfying also that the Shilov boundary of Gx;r is a singleton θ(x; r), and that the holomorphically convexe envelope 0 of θ(x; r) is Gx;r. We have Gx;0 = Gx, and for r > r we have Gx;r ⊂ Gx;r0 . 6= When r goes to +1, θ(x; r) goes to the neutral element. The set fθ(x; r) j r ≥ 0g is a segment joining θ(x) to the neutral element. 2 θ(x; 0) θ(x; r) This is an heuristic picture representing Gx and its ltrations. Pictorially, the group stabg (x) is the set of lower extremal points, Gx is the whole picture and Gx;r is the orange part. The k-points of Gx;r are the orange lower extremal points, they coincide with the corresponding Moy-Prasad group in many cases. The Shilov boundaries of Gx and Gx;r are the points θ(x; 0) and θ(x; r). The neutral element is the central lower point. Let us introduce some terminology. A k-anoid group H is a k-anoid Demazure group if it is the generic ber of the formal completion of a De- mazure model G of G along its special ber, it is a potentially k-anoid De- mazure group if there exists a K-anoid extension of k such that H ×M(k) an M(K) is a K-anoid Demazure group of G ×M(k) M(K); if the extension can be choosen nite then H is a rational potentially k-anoid Demazure group. We work with rational potentially k-anoid Demazure groups. Let us x such a H. Let us x a nite extension K=k such that H ×M(k) M(K) is the generic ber of the formal completion of a Demazure model G=K◦ of G ×k K along its special ber. We explain in this article that G is well- dened (unique) and Gal(K=k)-stable. Now let r be a positive rational number, we can assume r 2 ord(K). We dene k-anoid groups Hr taking the projection an of the generic ber of the prK=k : G ×M(k) M(K) !M(k) 3 formal completion of Γe(K;k)r(G) along its special ber. Here Γe(K;k)r(G) is the schematic e(K; k)r-th congruence of G [20], [13]. We show that Hr is a an well-dened k-anoid subgroup of G and that the Shilov boundary of Hr is a singleton σr and that the holomorphically convex envelope of σr is Hr. Now each rational point x in the Bruhat-Tits building gives rise to a rational potentially k-anoid Demazure groups Gx. Applying the general construc- tion, we get k-anoid ltrations Gx;r for any positive rational number. We show that this construction is compatible by base change in a natural sense and that in tame situations Gx;r(k) is the corresponding Moy-Prasad group for r > 0. We then study the map an θ : BT(G; k) × Q≥0 ! G ; where BT(G; k) denotes the dense subset of BT(G; k) made of rational points. We prove that it is continous and injective. We also dene simi- lar ltrations for the Lie algebra of G. An appendix compare our notion of rational points with Broussous-Lemaire one [4]. Notations p : a prime number k=Qp : a nite extension πk : a uniformizer of k ord : valuation on algebraic extensions of k such that ord(πk) = 1 e > 1 : a real number strictly bigger than 1 |•| = e−ord(•) norm ◦ k = fx 2 k j jxj ≤ 1 g = ok ◦◦ k = fx 2 k j jxj < 1 g = pk k~ = k◦=k◦◦ residual eld G : connected reductive k-group scheme BT(G; k): reduced Bruhat-Tits building BTE(G; k): enlarged Bruhat-Tits building Gan : analytication of G (Berkovich k-analytic space) 1 Schematic congruence subgroups In this section we recall some results about congruence subgroups for group schemes over k◦. Congruence groups are built using dilatations [3] [10]. It preserves group schemes structures. We refer the reader to [3], [20], [13] and [10] for dilatations, higher dilatations and schematic congruence groups (see also [17]). We now introduce directly congruence subgroups. 4 Denition 1.1. Let G = spec(A) be a at k◦-group scheme of nite type and n be a positive integer. Let I be the augmentation ideal of A. Put −n X −in i . Then is An = A[πk I] = A + πk I ⊂ A ⊗k◦ k Γn(G) := spec(An) i≥1 called the n-th congruence subgroup of G. It is a canonical group scheme satisfying ◦ ◦ ◦ n ◦ Γn(G)(k ) = ker (G(k ) ! G(k /πk k )) : ◦ This is a at k -group scheme together with a morphism Γn(G) ! G. Let X = spec(A) be an ane k-scheme of nite type. Let K=k be a nite Galois extension. Let X = spec(A) be an ane at K◦-scheme of nite type such that X×K◦ K = X ×k K: We thus have A⊗K◦ K = A⊗k K. By atness, we see A as a subset of A⊗k K. In this situation, we say that X is Gal(K=k)- stable if A is Gal(K=k)-stable in A ⊗k K. The following Proposition shows that Galois stability is preserved under the operations of taking congruence subgroups.
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