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Buildings and Berkovich Spaces Buildings and Berkovich Spaces Annette Werner Goethe-Universit¨at, Frankfurt am Main DMV-Jahrestagung Munchen¨ 2010 Emmy Noether Lecture 1 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 This talk reports on joint work with Amaury Thuillier and Bertrand R´emy(Lyon). Our results generalize results of Vladimir Berkovich who investigated the case of split groups. 2 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Non-archimedean fields K non-Archimedean field, i.e. K is complete with respect to a non-trivial absolute value j jK satisfying ja + bjK 5 maxfjajK ; jbjK g: ∗ ∗ K is called discrete if the value group jK j ⊂ R is discrete. Non-archimedean analysis has special charms: 1 X an converges if and only if an ! 0: n=1 3 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Non-archimedean fields Examples: K = k(T ) formal Laurent series over any ground field k with j P a T nj = e−n0 if a 6= 0 n≥n0 n n0 K = CffT gg Puiseux series −vp(x) K = Qp, the completion of Q with respect to jxj = p algebraic extensions of Qp K = Cp, the completion of the algebraic closure of Qp 4 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Semisimple groups G semisimple group over K, i.e. G ,! GLn;K closed algebraic subgroup such that rad(G)(= biggest connected solvable normal subgroup) = 1 Examples: SLn; PGLn; Sp2n; SOn over K SLn(D) D central division algebra over K 5 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Main result Goal: Embed the Bruhat-Tits building B(G; K) associated to G in the Berkovich analytic space G an associated to G. Hope: Investigate the building with the help of the ambiant Berkovich space G an. 6 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Bruhat-Tits buildings: Example Archimedean Example: G = SL(2; R) H = SO(2; R) maximal compact subgroup G=H = H = fz 2 C : Im(z) > 0g upper half-plane Non-Archimedean analog: p prime number G = SL(2; Qp) H = SL(2; Zp) maximal compact subgroup. G=H is a totally disconnected topological space. 7 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Bruhat-Tits buildings: Example 2 Note: H = fnorms on R g=scaling. Goldman-Iwahori: 2 B(SL2; Qp) = fNon-archimedean norms on Qpg= scaling Topology of pointwise convergence SL(2; Qp)−action a Stabilizer of the norm b 7! maxfjaj; jbjg is the maximal compact subgroup SL(2; Zp) 8 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Bruhat-Tits buildings: Example B(SL2; Qp) is an infinite (p + 1)−valent tree: (p = 2): 9 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Bruhat-Tits buildings In general: The building B(G; K) is obtained by glueing real vector spaces (apartments). r Every maximal split torus T ⊂ G, i.e. T ' Gm;K , induces an apartment A(T ), which is defined as the real cocharacter space A(T ) = HomK (Gm; T ) ⊗Z R. The glueing process is defined with deep (and quite technical) results by Bruhat and Tits. B(G; K) is a complete metric space with a continuous G(K)−action. If K is discrete, B(G; K) carries a (poly-)simplicial structure. 10 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Apartment for Sp4 11 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Apartment for PGL3 12 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Some part of B(PGL3; Qp) P. Garrett: Buildings and classical groups 13 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Why are Bruhat-Tits buildings useful? B(G; K) is a nice space on which G(K) acts Cohomology of arithmetic groups (Borel-Serre) B(G; K) encodes information about the compact subgroups of G(K) Representation theory of G(K) (Schneider-Stuhler) Bruhat-Tits buildings are non-Archimedean analogs of Riemann symmetric spaces of non-compact type Buildings can be used to prove results for symmetric spaces (e.g. Kleiner-Leeb) 14 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Berkovich Spaces A Berkovich space is a non-Archimedean analytic space with good topological properties. Archimedean case: X smooth projective variety over C. Then X (C) is a complex projective manifold. Non-archimedean case: X smooth projective variety over K. Then X (K) inherits a non-Archimedean topology from K with bad topological properties, e.g. it is totally disconnected. 15 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Berkovich Spaces Tate, Raynaud... Define non-Archimedean analytic functions by a suitable Grothendieck topology Berkovich Enlarge X (K) to a topological space X an with good properties. 16 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Berkovich unit disc Example: The Berkovich unit disc Assume for simplicity that K is algebraically closed. P n A = Kfzg = f formal series f (z) = n≥0 anz with an ! 0g k f k= maxjanjK Gauss norm on A n 17 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Berkovich unit disc M(A) = f bounded multiplicative seminorms on A extending j jK g is the Berkovich unit disc. Hence every γ 2 M(A) is a function γ : A ! R=0 satisfying γjK = j jK γ(fg) = γ(f ) γ(g) γ(f + g) 5 γ(f ) + γ(g) γ ≤ c jj jj Every a 2 K with jajK 5 1 induces a point jf ja = jf (a)jK in M(A) The Gauss norm is multiplicative, i.e. a point in M(A). 18 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Non-archimedean balls The other seminorms in M(A) can be described with closed non-Archimedean discs D(a; r) = fx 2 K : jx − aj 5 rg Note: Two non-Archimedean closed discs are either disjoint or nested 19 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Berkovich unit disc Basic fact: The Gauss norm is the supremum norm on D(0; 1): The Berkovich unit disc consists of the following points: Points of type 1: jf ja = jf (a)jK for a 2 D(0; 1): Points of type 2: jf ja;r = sup jf (x)jK for D(a; r) ⊂ D(0; 1) x2D(a;r) and r 2 jK ∗j Points of type 3: jf ja;r = sup jf (x)j for D(a; r) ⊂ D(0; 1) x2D(a;r) and r 2= jK ∗j Points of type 4: jf ja;r = lim jf ja ;r for a nested sequence n!1 n n D(a1; r1) ⊃ D(a2; r2) ::: of closed discs in D(0; 1) 20 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Berkovich unit disc (from J.H. Silverman: The arithmetic of dynamical systems) 21 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Berkovich unit disc Endow the Berkovich unit disc M(A) with the topology of pointwise convergence of seminorms evaluated on A. Then M(A) is a compact, uniquely path-connected Hausdorff space containing fx 2 K : jxjK 5 1g as a dense subspace. 22 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Berkovich spaces Similary one can define Berkovich discs of any radius r > 0. Berkovich affine line: 1 an (AK ) = union of all Berkovich discs of positive radius = fmultiplicative seminorms on K[z]g: Berkovich projective line: 1 an (PK ) can be constructed by glueing two Berkovich unit discs. 23 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Analytification of algebraic varieties In general: X = Spec A for A = K[x1;:::; xn]=a Berkovich space X an corresponding to X : an X = fmultiplicative seminorms on A extending j jK g An analogous definition over the complex numbers yields X (C) by a theorem of Gelfand-Mazur. 24 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Berkovich spaces Berkovich spaces have found a variety of applications, e.g. to prove a conjecture of Deligne on vanishing cycles (Berkovich) in local Langlands theory (Harris-Taylor) to develop a p−adic avatar of Grothendieck's \dessins d'enfants" (Andr´e) to develop a p−adic integration theory over genuine paths (Berkovich) for p−adic harmonic analysis and p−adic dynamics with applications in Arakelov Theory (Baker, Chambert-Loir, Rumely, Thuillier,...) 25 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Embedding Theorem G semisimple algebraic group over K G an Berkovich space associated to G We define a continuous, G(K)− equivariant embedding ν : B(G; K) −! G an using the following theorem: 26 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Embedding Theorem Theorem i) For all x 2 B(G; K) there exists an (affinoid) subgroup an Gx = M(Ax ) ⊂ G such that Gx (L) = StabG(L)(x) for all non-Archimedean fields L ⊃ K: an ii) Gx has a unique maximal point in G (Shilov boundary point), i.e. there exists a unique ν(x) 2 G an such that all f 2 Ax achieve their maximum on ν(x): 27 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Embedding Theorem Tools: Bruhat-Tits theory, Berkovich's characterization of Shilov boundary points, descent theory for affinoids New idea: Any point x becomes special after base extension with a suitable L=K: 28 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Embedding Theorem: Example Example G = SL2 T ⊂ G torus of diagonal matrices a 0 A(T ) = Hom( ; T ) ⊗ = µ for µ : a 7! : Gm Z R R 0 a−1 ( ) ( ) 1 0 1 v U = : u 2 K U = : v 2 K − u 1 + 0 1 Ω = U−TU+ ⊂ SL2 big cell Note that Ω = Spec K[a; a−1; u; v]: 29 / 33 Buildings and Berkovich Spaces DMV-Jahrestagung Munchen¨ 2010 Embedding Theorem: Example The embedding ν is constructed apartment-wise.
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