, G x,r is a k-anoid subgroup of G an
, The Shilov boundary of G x,r is reduced to a point that we denote ?(x, r)
, G x,r is the holomorphically convex envelope of ?(x, r)
, The k-anoid algebra of G x,r is the completion of Hopf(G) relatively to the norm | | Hopf(? e(K,k)r (G)) | Hopf(G)
, Let x be a rational point in the reduced Bruhat-Tits building of G, let r be a positive rational number and let g ? G(k), then 1. G g.x,r = gG x
, G i k /k (x),r = G x,r × M(k) M(k )
, G i k /k (x),r (k ) ? G(k) = G x,r (k)
, The map ? : BT R rat (G, k) × Q ?0 ? G an is continuous and injective
, We have to show that g.?(x, r) = ?(g.(x, r)). This is a direct consequence of 2.5.4, indeed ?(g.(x, r)) = ?(g.x, r) = g?(x, r)g ?1 = g
, Let K/k be the extension used to dene G x,r as G x,r = pr K/k ( ? e(K,k)r (G) ? ), we can assume that k ? K. We have G x,r = pr K/k ( ? e(K,k)r (G) ? ), thus G x,r = pr k /k (G x,r )
G i K/k (x),r by denition of G x,r and since K/k is a Galois extension ,
, K/k (pr ?1 k /k (G x,r )) = G i K/k (x),r . We also have pr ?1
,
/k (x),r by denition of G x,r and since K/k is a Galois extension ,
, K/k (pr ?1 k /k (G x,r )) = pr ?1
,
, The point x is a (rational) special point of BT R (G, k) and i K/k (x) ? BT R (G, K) is special for any nite extension K/k. The group G is not split over k, it is split over l. Let us make explicit the group G x,0 . We need to nd an extension K/k such that G is split over K, i K/k (x) is special, and r = 0 ? ord(K), shows that G × spec(k) spec(l) G m /l. The reduced Bruhat-Tits building BT R (G, k) is a singleton {x}
, ? inf ??l × {|?| | f ? ?(Hopf(G) ? 1) ? Hopf(G × k l)}
,
,
, And
,
X 2 ? 2Y 2 ? 1 relatively to this norm, in order to simplify notation let us put || || = | | Hopf(G) | Hopf(G) ,
,
,
, Here we do not use a torus, the approach is algebraic. We use the explicit description of Hopf algebras of congruence groups of G. For any k ? -group scheme G, we use that Lie(G) is explicitely given by Hom k ? ?mod (I/I 2 , k ? ) where I, the situation of reductive group and it is constructed using a maximal torus, roots groups and splitting the reductive group
, Let k be a non arch. local eld, and let ? , k ? , the usual associated notations
, Let G = spec(A) be an ane smooth (thus at) k ? -group scheme. Let A be the k ? -Hopf algebra of G. Let ? : A ? k ? be the counit. Let I := ker(?) be the augmentation ideal. Let I 2 be the ideal II
, I/I 2 is a free k ? -module
, n of I/I 2 ; and choose alsog 1
, Let G = spec(A) be a at ane k ? -scheme satisfying the hypothesis of [32, Lemma 5.1]. Then 1. A contains no non-zero k ? -divisble element. 2. The ideal of augmentation I and its square power I 2 contain no nonzero k ? -divisible element
, A, I and I 2 are free k ? -modules
, Let r ? s, then 1. Let x ? A r , then there exists a positive integer N such that ? N x ? A s . 2. Let x ? I r , then there exists a positive integer N such that ? N x ? I s 3. Let x ? I r /I r 2 , then ? r?s x ? I s /I s 2
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