Editing Building (mathematics) (section)

==Spherical and affine buildings for {{math|SL<sub>''n''</sub>}}==
The simplicial structure of the affine and spherical buildings associated to {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}, as well as their interconnections, are easy to explain directly using only concepts from elementary [[algebra]] and [[geometry]] (see {{harvnb|Garrett|1997}}). In this case there are three different buildings, two spherical and one affine. Each is a union of ''apartments'', themselves simplicial complexes. For the affine building, an apartment is a simplicial complex [[tessellation|tessellating]] Euclidean space {{math|'''E'''<sup>''n''−1</sup>}} by {{math|(''n'' − 1)}}-dimensional simplices; while for a spherical building it is the finite simplicial complex formed by all {{math|(''n'' − 1)!}} simplices with a given common vertex in the analogous tessellation in {{math|'''E'''<sup>''n''−2</sup>}}.

Each building is a simplicial complex {{mvar|X}} which has to satisfy the following axioms:

* {{mvar|X}} is a union of apartments.
* Any two simplices in {{mvar|X}} are contained in a common apartment.
* If a simplex is contained in two apartments, there is a simplicial isomorphism of one onto the other fixing all common points.

===Spherical building===
Let {{mvar|F}} be a [[field (mathematics)|field]] and let {{mvar|X}} be the simplicial complex with vertices the non-trivial vector subspaces of {{math|''V'' {{=}} ''F''<sup>''n''</sup>}}. Two subspaces {{math|''U''<sub>1</sub>}} and {{math|''U''<sub>2</sub>}} are connected if one of them is a subset of the other. The {{mvar|k}}-simplices of {{mvar|X}} are formed by sets of {{math|''k'' + 1}} mutually connected subspaces. Maximal connectivity is obtained by taking {{math|''n'' − 1}} proper non-trivial subspaces and the corresponding {{math|(''n'' − 1)}}-simplex corresponds to a ''[[Flag (linear algebra)|complete flag]]''

: {{math|(0) ⊂ ''U''<sub>1</sub> ⊂ ··· ⊂ ''U''<sub>''n'' – 1 </sub> ⊂ ''V''}}

Lower dimensional simplices correspond to partial flags with fewer intermediary subspaces {{math|''U''<sub>''i''</sub>}}.

To define the apartments in {{mvar|X}}, it is convenient to define a ''frame'' in {{mvar|V}} as a basis ({{math|''v''<sub>''i''</sub>}}) determined up to scalar multiplication of each of its vectors {{math|''v''<sub>''i''</sub>}}; in other words a frame is a set of one-dimensional subspaces {{math|''L''<sub>''i''</sub> {{=}} ''F''·''v''<sub>''i''</sub>}} such that any {{mvar|k}} of them generate a {{mvar|k}}-dimensional subspace. Now an ordered frame {{math|''L''<sub>1</sub>, ..., ''L''<sub>''n''</sub>}} defines a complete flag via

: {{math|''U''<sub>''i''</sub> {{=}} ''L''<sub>1</sub> ⊕ ··· ⊕ ''L''<sub>''i''</sub>}}

Since reorderings of the various {{math|''L''<sub>''i''</sub>}} also give a frame, it is straightforward to see that the subspaces, obtained as sums of the {{math|''L''<sub>''i''</sub>}}, form a simplicial complex of the type required for an apartment of a spherical building. The axioms for a building can easily be verified using the classical [[Schreier refinement theorem|Schreier refinement argument]] used to prove the uniqueness of the [[Jordan–Hölder decomposition]].

===Affine building===
Let {{mvar|K}} be a field lying between {{math|'''Q'''}} and its [[p-adic number|{{mvar|p}}-adic completion]] {{math|'''Q'''<sub>''p''</sub>}} with respect to the usual [[Archimedean property|non-Archimedean]] [[p-adic norm|{{mvar|p}}-adic norm]] {{math|{{norm|''x''}}<sub>''p''</sub>}} on {{math|'''Q'''}} for some prime {{mvar|p}}. Let {{mvar|R}} be the [[subring]] of {{mvar|K}} defined by

:{{math|''R'' {{=}} { ''x'' : {{norm|''x''}}<sub>''p''</sub> ≤ 1 } }}

When {{math|''K'' {{=}} '''Q'''}}, {{mvar|R}} is the [[Localization of a ring|localization]] of {{math|'''Z'''}} at {{mvar|p}} and, when {{math|''K'' {{=}} '''Q'''<sub>''p''</sub>}}, {{math|''R'' {{=}} '''Z'''<sub>''p''</sub>}}, the [[p-adic integer|{{mvar|p}}-adic integers]], i.e. the closure of {{math|'''Z'''}} in {{math|'''Q'''<sub>''p''</sub>}}.

The vertices of the building {{mvar|X}} are the {{mvar|R}}-lattices in {{math|''V'' {{=}} ''K''<sup>''n''</sup>}}, i.e. {{mvar|R}}-[[submodules]] of the form

:{{math|''L'' {{=}} ''R''·''v''<sub>1</sub> ⊕ ··· ⊕  ''R''·''v''<sub>''n''</sub>}}

where {{math|(''v''<sub>''i''</sub>)}} is a basis of {{mvar|V}} over {{mvar|K}}. Two lattices are said to be ''equivalent'' if one is a scalar multiple of the other by an element of the multiplicative group {{math|''K''*}} of {{mvar|K}} (in fact only integer powers of {{mvar|p}} need be used). Two lattices {{math|''L''<sub>1</sub>}} and {{math|''L''<sub>2</sub>}} are said to be ''adjacent'' if some lattice equivalent to {{math|''L''<sub>2</sub>}} lies between {{math|''L''<sub>1</sub>}} and its sublattice {{math|''p''·''L''<sub>1</sub>}}: this relation is symmetric. The {{mvar|k}}-simplices of {{mvar|X}} are equivalence classes of {{math|''k'' + 1}} mutually adjacent lattices, The {{math|(''n'' − 1)}}-simplices correspond, after relabelling, to chains

:{{math|''p''·''L''<sub>''n''</sub> ⊂ ''L''<sub>1</sub> ⊂ ''L''<sub>2</sub> ⊂ ··· ⊂ ''L''<sub>''n'' – 1 </sub> ⊂ ''L''<sub>''n''</sub>}}

where each successive quotient has order {{mvar|p}}. Apartments are defined by fixing a basis {{math|(''v''<sub>''i''</sub>)}} of {{mvar|V}} and taking all lattices with basis {{math|(''p''<sup>''a''<sub>''i''</sub></sup> ''v''<sub>''i''</sub>)}} where {{math|(''a''<sub>''i''</sub>)}} lies in {{math|'''Z'''<sup>''n''</sup>}} and is uniquely determined up to addition of the same integer to each entry.

By definition each apartment has the required form and their union is the whole of {{mvar|X}}. The second axiom follows by a variant of the Schreier refinement argument. The last axiom follows by a simple counting argument based on the orders of finite Abelian groups of the form

:{{math|''L'' + ''p''<sup>''k''</sup> ·''L''<sub>''i''</sub> / ''p''<sup>''k''</sup> ·''L''<sub>''i''</sub>}}

A standard compactness argument shows that {{mvar|X}} is in fact independent of the choice of {{mvar|K}}. In particular taking {{math|''K'' {{=}} '''Q'''}}, it follows that {{mvar|X}} is countable. On the other hand, taking {{math|''K'' {{=}} '''Q'''<sub>''p''</sub>}}, the definition shows that {{math|GL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}} admits a natural simplicial action on the building.

The building comes equipped with a ''labelling'' of its vertices with values in {{math|'''Z''' / ''n'''''Z'''}}. Indeed, fixing a reference lattice {{mvar|L}}, the label of {{mvar|M}} is given by

:{{math|label(''M'') {{=}} log<sub>''p''</sub> {{abs|''M'' / ''p''<sup>''k''</sup> ''L''}} modulo ''n''}}

for {{mvar|k}} sufficiently large. The vertices of any {{math|(''n'' – 1)}}-simplex in {{mvar|X}} has distinct labels, running through the whole of {{math|'''Z''' / ''n'''''Z'''}}. Any simplicial automorphism {{mvar|φ}} of {{mvar|X}} defines a permutation {{mvar|π}} of {{math|'''Z''' / ''n'''''Z'''}} such that {{math|label(''φ''(''M'')) {{=}} ''π''(label(''M''))}}. In particular for {{mvar|g}} in {{math|GL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}},

:{{math|label(''g''·''M'') {{=}} label(''M'') + log<sub>''p''</sub> {{norm|det ''g''}}<sub>''p''</sub> modulo ''n''}}.

Thus {{mvar|g}} preserves labels if {{mvar|g}} lies in {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}.

===Automorphisms===
Tits proved that any label-preserving [[automorphism]] of the affine building arises from an element of {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}. Since automorphisms of the building permute the labels, there is a natural homomorphism

:{{math|Aut ''X'' → ''S''<sub>''n''</sub>}}.

The action of {{math|GL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}} gives rise to an [[cyclic permutation|{{mvar|n}}-cycle]]&nbsp;{{mvar|τ}}. Other automorphisms of the building arise from [[outer automorphism]]s of {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}} associated with automorphisms of the [[Dynkin diagram]]. Taking the standard symmetric bilinear form with orthonormal basis {{math|''v''<sub>''i''</sub>}}, the map sending a lattice to its dual lattice gives an automorphism whose square is the identity, giving the permutation {{mvar|σ}} that sends each label to its negative modulo {{mvar|n}}. The image of the above homomorphism is generated by {{mvar|σ}} and {{mvar|τ}} and is isomorphic to the [[dihedral group]] {{math|''D''<sub>''n''</sub>}} of order {{math|2''n''}}; when {{math|''n'' {{=}} 3}}, it gives the whole of {{math|''S''<sub>3</sub>}}.

If {{mvar|E}} is a finite [[Galois extension]] of {{math|'''Q'''<sub>''p''</sub>}} and the building is constructed from {{math|SL<sub>''n''</sub>(''E'')}} instead of {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}, the [[Galois group]] {{math|Gal(''E'' / '''Q'''<sub>''p''</sub>)}} will also act by automorphisms on the building.

===Geometric relations===
Spherical buildings arise in two quite different ways in connection with the affine building {{mvar|X}} for {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}:

* The [[link (geometry)|link]] of each vertex {{mvar|L}} in the affine building corresponds to submodules of {{math|''L'' / ''p''·''L''}} under the finite field {{math|''F'' {{=}} ''R'' / ''p''·''R'' {{=}} '''Z''' / (''p'')}}. This is just the spherical building for {{math|SL<sub>''n''</sub>(''F'')}}.
* The building {{mvar|X}} can be ''[[compactification (mathematics)|compactified]]'' by adding the spherical building for {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}} as boundary "at infinity" (see {{harvnb|Garrett|1997}} or {{harvnb|Brown|1989}}).

===Bruhat–Tits trees with complex multiplication===

When {{mvar|L}} is an archimedean local field then on the building for the group {{math|SL<sub>2</sub>(''L'')}} an additional structure can be imposed of a building with complex multiplication. These were first introduced by [[Martin L. Brown]]  ({{harvnb|Brown|2004}}). These buildings arise when a quadratic extension of {{mvar|L}} acts on the vector space {{math|''L''<sup>2</sup>}}. These building with complex multiplication can be extended to any global field. They  describe  the action of the Hecke operators on  Heegner points on the classical modular curve {{math|''X''<sub>0</sub>(''N'')}} as well as on the Drinfeld modular curve {{math|''X''{{su|b=0|p=Drin}}(''I'')}}. These buildings with complex multiplication are completely classified for the case of {{math|SL<sub>2</sub>(''L'')}} in {{harvnb|Brown|2004}}